# Analysis of a high-order unfitted finite element method for elliptic interface problems

Analysis of a high-order unfitted finite element method for elliptic interface problems Abstract In the context of unfitted finite element discretizations, the realization of high-order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method that achieves a high-order approximation of the geometry for domains that are implicitly described by smooth-level set functions. The method is based on a parametric mapping, which transforms a piecewise planar interface reconstruction to a high-order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping, are easy to implement. In this article, we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high-order discretization. The theoretical results are confirmed in numerical experiments. 1. Introduction 1.1 Motivation In recent years, there has been a strong increase in the research on development and analysis of unfitted finite element methods (FEMS). In unfitted FEMs, a geometry description, which is separated from the computational mesh, is used to provide a more flexible handling of the geometry compared with traditional conforming mesh descriptions. Often a level set function is used to describe geometries implicitly. Recently, significant progress has been made in the construction, analysis and application of methods of this kind (see, for instance Hansbo & Hansbo, 2002; Groß & Reusken, 2007; Olshanskii et al., 2009; Fries & Belytschko, 2010; Burman & Hansbo, 2012; Burman et al., 2014). Despite these achievements, the development and rigorous error analysis of high-order accurate unfitted FEMs is still challenging. This is mainly due to the fact that efficient, highly accurate numerical integration on domains that are implicitly described is not straight forward. In the recent article by Lehrenfeld (2016a), a new approach based on isoparametric mappings of the underlying mesh (outlined in Section 1.4 below) has been introduced, which allows for an efficient, robust and highly accurate numerical integration on domains that are described implicitly by a level set function. The main contribution of this article is an error analysis of this method applied to a model interface problem. 1.2 Literature We review the state of the art with respect to higher-order geometry approximations in the literature to put the approach proposed in Lehrenfeld (2016a) and analysed in this article into its context. For discretizations based on piecewise linear (unfitted) finite elements, a numerical integration approach which is second order accurate suffices to preserve the overall order of accuracy. This is why an approximation with piecewise planar geometries is an established approach. Piecewise planar approximations allow for a tesselation into simpler geometries, e.g., simplices, on which standard quadrature rules are applicable. Hence, a robust realization is fairly simple. It can be applied to quadrilateral and hexahedral meshes, as in the marching cube algorithm (Lorensen & Cline, 1987) as well as to simplex meshes, cf. (among others) Lehrenfeld (2015b, Chapter 4), Mayer et al. (2009), Nærland (2014, Chapter 5) for triangles and tetrahedra and Lehrenfeld (2015a), Lehrenfeld (2015b, Chapter 4) for 4-prisms and pentatopes (4-simplices). Such strategies are used in many simulation codes, e.g. (Engwer & Heimann, 2012; Burman et al., 2014; Renard & Pommier, 2014; Groß et al., 2002; Carraro & Wetterauer, 2016) and are often combined with a geometrical refinement in the quadrature. Especially on octree-based meshes, this can be done very efficiently (Chernyshenko & Olshanskii, 2015). However, by construction, the tesselation approach is only second-order accurate, and hence limits the overall accuracy when combined with higher-order finite elements. Many unfitted discretizations based on piecewise linear finite elements have a natural extension to higher-order finite element spaces, see, for instance (Parvizian et al., 2007; Bastian & Engwer, 2009; Massjung, 2012; Johansson & Larson, 2013). It requires, however, additional new techniques to obtain also higher-order accuracy when errors due to the geometry approximation and quadrature are taken into account. We review such techniques, starting with approaches from the engineering literature. One such technique is based on moment-fitting approaches, resulting in special quadrature rules (Müller et al., 2013; Sudhakar & Wall, 2013). This approach provides (arbitrary) high-order accurate integration on implicit domains. However, the construction of these quadrature rules is fairly involved and the positiveness of quadrature weights cannot be guaranteed, which can lead to stability problems. Furthermore, as far as we know, there is no rigorous complexity and error analysis. For special cases also other techniques have been proposed in the literature to obtain high-order accurate approximations of integrals on unfitted domains. In Saye (2015), a new algorithm is presented, which can achieve arbitrary high-order accuracy and guarantee positivity of integration weights. However, it is applicable only on tensor-product elements. The approach is based on the idea of locally interpreting the interface as a graph over a hyperplane. In the community of the extended FEM (XFEM) approaches, applying a parametric mapping of the sub-triangulation are often used (e.g., Cheng & Fries, 2010; Dréau et al., 2010; Fries & Omerovi, 2015). The realization of such strategies is technically involved, especially in three (or higher) dimensions. Moreover, ensuring robustness (theoretically and practically) of these approaches is difficult. Concerning the finite cell method (Parvizian et al., 2007), which is also a (high-order) unfitted FEM, we refer to Abedian et al. (2013) for a comparison of different approaches for the numerical integration on unfitted geometries. We note that the aforementioned articles on numerical integration in higher-order unfitted FEMs, which are all from the engineering literature, only present new approaches for the numerical integration and carry out numerical convergence studies. Rigorous error analyses of these methods are not known. We now discuss techniques and corresponding rigorous error analyses that have been considered in the mathematical literature. In the recent articles (Burman et al., 2015; Boiveau et al., 2016), the piecewise planar approximation discussed before has been used and combined with a correction in the imposition of boundary values (based on Taylor expansions), to allow for higher-order accuracy in fictitious domain methods applied to the Dirichlet problems. Optimal order a priori error bounds are derived. For a higher-order unfitted discretization of partial differential equations (PDEs) on surfaces, a parametric mapping of a piecewise planar interface approximation has been developed and analysed in Grande & Reusken (2016). These are the only articles, which we know of, in which geometry errors in higher-order unfitted FEMs are included in the error analysis. We also mention relevant numerical analysis articles using low-order unfitted or higher-order geometrically conforming (‘fitted’) methods. For unfitted piecewise linear finite element discretizations of PDEs on surfaces, errors caused by geometry approximation are analysed in Deckelnick et al. (2014); Burman et al. (2016). Error analyses for fitted isoparametric finite elements for the accurate approximation of curved boundaries have been developed in the classical contributions Lenoir (1986); Bernardi (1989); Ciarlet (1991). In the context of interface problems, surface-bulk coupling isoparametric-fitted techniques are analysed in Elliott & Ranner (2013). Recently, these isoparametric fitted methods have also been used to adjust a simple background mesh to curved implicitly defined geometries in a higher-order accurate fashion (Demlow, 2009; Basting & Weismann, 2013; Gawlik & Lew, 2014; Omerović & Fries, 2017). The approach analysed in this article is similar to the above-mentioned approaches Lenoir (1986); Cheng & Fries (2010); Dréau et al. (2010); Basting & Weismann (2013); Gawlik & Lew (2014); Grande & Reusken (2016); Omerović & Fries (2017) in that it is also based on a piecewise planar geometry approximation, which is significantly improved using a parametric mapping. The important difference compared with Lenoir (1986); Basting & Weismann (2013); Gawlik & Lew (2014); Omerović & Fries (2017) is that we consider an unfitted discretization and compared with Cheng & Fries (2010); Dréau et al. (2010); Grande & Reusken (2016) is that we consider a parametric mapping of the underlying mesh rather than the sub-triangulation or only the interface. The discretization on the higher-order geometry approximation that we obtain from a parametric mapping is based on a variant of Nitsche’s method (Nitsche, 1971) for the imposition of interface conditions in nonconforming unfitted finite element spaces (Hansbo & Hansbo, 2002). To our knowledge, there is no literature in which geometrical errors for isoparametric or higher-order unfitted Nitsche-type discretizations have been considered. 1.3 The problem setting On a bounded connected polygonal domain $${\it{\Omega}} \subset \Bbb{R}^d$$, $$d=2,3$$, we consider the model interface problem \begin{align} - \mathrm{div} (\alpha_i \nabla {u}) &= \, f_i \quad \text{in} \,\, {\it{\Omega}}_i, \, i=1,2, \label{eq:ellmodel1} \\ \end{align} (1.1a) \begin{align} [\hspace{-2pt}[{{\alpha} \nabla {u}}]\hspace{-2pt}]_{{\it{\Gamma}}} \cdot n_{\it{\Gamma}} &= \, 0, \quad [\hspace{-2pt}[{{u}}]\hspace{-2pt}]_{{\it{\Gamma}}} = 0 \quad \text{on} \,\, {\it{\Gamma}}, \label{eq:ellmodel2} \\ \end{align} (1.1b) \begin{align} u &= 0 \quad \text{on} \,\, \partial \,\, {\it{\Omega}}. \end{align} (1.1c) Here, $${\it{\Omega}}_1 \cup {\it{\Omega}}_2= {\it{\Omega}}$$ is a nonoverlapping partitioning of the domain, $${\it{\Gamma}} = \bar {\it{\Omega}}_1 \cap \bar {\it{\Omega}}_2$$ is the interface and $$[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]_{{\it{\Gamma}}}$$ denotes the usual jump operator across $${\it{\Gamma}}$$. $$f_i$$, $$i=1,2$$ are domainwise described sources. In the remainder, we will also use the source term $$f$$ on $${\it{\Omega}}$$, which we define as $$f|_{{\it{\Omega}}_i} = f_i$$, $$i=1,2$$. The diffusion coefficient $$\alpha$$ is assumed to be piecewise constant, i.e., it has a constant value $$\alpha_i$$ on each subdomain $${\it{\Omega}}_i$$. We assume simplicial triangulations of $${\it{\Omega}}$$ which are not fitted to $${\it{\Gamma}}$$. Furthermore, the interface is characterized as the zero level of a given level set function $$\phi$$. 1.4 Basic idea of the isoparametric unfitted discretization The new idea of the method introduced in Lehrenfeld (2016a) is to construct a parametric mapping of the underlying triangulation based on a higher-order (i.e., degree at least 2) finite element approximation $$\phi_h$$ of the level set function $$\phi$$, which characterizes the interface. This mapping, which is easy to construct, uses information extracted from the level set function to map a piecewise planar interface approximation of the interface to a higher-order approximation, cf. the sketch in Fig. 1. Fig. 1. View largeDownload slide Basic idea of the method in Lehrenfeld (2016a): The geometry description with the level set function approximation $$\phi_h$$ is highly accurate but implicit (left). The zero level $${\it{\Gamma}}^{\text{lin}}$$ of the piecewise linear interpolation $$\hat{\phi}_h$$ has an explicit representation, but is only second order accurate (center). $${\it{\Gamma}}^{\text{lin}}$$ is mapped approximately to the implicit interface $$\{\phi_h=0\}$$ using a mesh transformation $${\it{\Theta}}_h$$, resulting in a highly accurate and explicit representation $${\it{\Gamma}}_h = {\it{\Gamma}}^{\text{lin}} \circ {\it{\Theta}}_h^{-1}$$ (right). Fig. 1. View largeDownload slide Basic idea of the method in Lehrenfeld (2016a): The geometry description with the level set function approximation $$\phi_h$$ is highly accurate but implicit (left). The zero level $${\it{\Gamma}}^{\text{lin}}$$ of the piecewise linear interpolation $$\hat{\phi}_h$$ has an explicit representation, but is only second order accurate (center). $${\it{\Gamma}}^{\text{lin}}$$ is mapped approximately to the implicit interface $$\{\phi_h=0\}$$ using a mesh transformation $${\it{\Theta}}_h$$, resulting in a highly accurate and explicit representation $${\it{\Gamma}}_h = {\it{\Gamma}}^{\text{lin}} \circ {\it{\Theta}}_h^{-1}$$ (right). A higher-order isoparametric unfitted FEM is then obtained by first formulating a higher-order discretization with respect to the low-order geometry approximation, which serves as a reference configuration. The parametric mapping is then applied to improve the accuracy of the geometry approximation. The application of the transformation also induces a change of the finite element space, which renders the space an isoparametric finite element space. This approach is geometry based and can be applied to unfitted interface or boundary value problems as well as to PDEs on surfaces. All volume and interface integrals that occur in the implementation of the method can be formulated in terms of integrals on the reference configuration, i.e., on simplices cut by the piecewise planar approximation of the interface. Hence, quadrature is straightforward. For the definition of the discretization of the model problem (1.1) on the reference geometry, we take an unfitted higher-order ‘cut’ finite element space, where the cut occurs at the piecewise planar zero level of the piecewise linear finite element approximation of $$\phi_h$$. Applying the mesh transformation to this space induces an isoparametric unfitted finite element space for the discretization of (1.1). In the same way, as in the seminal article (Hansbo & Hansbo, 2002), the continuity of the discrete solution across the (numerical) interface is enforced in a weak sense using Nitsche’s method. The resulting discrete problem has a unique solution $$u_h$$ in the isoparametric unfitted finite element space ($$h$$ is related in the usual way to the size of the simplices in the triangulation). 1.5 Content and structure of the article In this article, we present a rigorous error analysis of the method introduced in Lehrenfeld (2016a), applied to the interface problem (1.1). The main result of the article is the discretization error bound in Theorem 5.12. As a corollary of that result, we obtain for isoparametric unfitted finite elements of degree $$k \geq 1$$ an error bound of the form $h^{-\frac12} \Vert [\hspace{-2pt}[{\tilde{u}_h}]\hspace{-2pt}]_{\it{\Gamma}} \Vert_{L^2({\it{\Gamma}})} + \|u- \tilde{u}_h \|_{H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \leq c h^k\big(\|u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} + \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \big),$ with a constant $$c>0$$ that is independent of $$h$$ and the interface position within the computational mesh. $$\tilde{u}_h := u_h \circ {\it{\Phi}}_h^{-1}$$ is the mapped discrete solution, where $${\it{\Phi}}_h$$ is a smooth transformation, which maps from the approximated to the exact domains $${\it{\Omega}}_i, i=1,2$$ and is close to the identity (precise definition given in Section 5.2). Here, $$\|\cdot\|_{H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}=\|\cdot\|_{H^{1,2}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$ and $$\|\cdot\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$ are broken Sobolev norms with $$\|\cdot\|_{W({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}^2 = \sum_{i=1,2} \|\cdot\|_{W({\it{\Omega}}_i)}^2$$, $$W \in \{H^1, H^{k+1}, H^{1,\infty} \}$$. As far as we know, this is the first rigorous higher-order error bound for an unfitted FEM applied to a problem with an implicitly given interface. The article is organized as follows. In Section 2, we introduce notation and assumptions, and define a finite element projection operator. The parametric mapping $${\it{\Theta}}_h$$ is introduced in Section 3. In Section 4, the isoparametric unfitted FEM is presented and numerical examples are given. The main contribution of this article is the error analysis of the method given in Section 5. 2. Preliminaries 2.1 Notation and assumptions We introduce notation and assumptions. The simplicial triangulation of $${\it{\Omega}}$$ is denoted by $${\mathcal T}$$ and the standard finite element space of continuous piecewise polynomials up to degree $$k$$ by $$V_h^k$$. The nodal interpolation operator in $$V_h^k$$ is denoted by $$I_k$$. For the ease of presentation, we assume quasi-uniformity of the mesh, s.t. $$h$$ denotes a characteristic mesh size with $$h\sim h_T:={\rm diam}(T)$$, $$T\in{\mathcal T}$$. We assume that the smooth interface $${\it{\Gamma}}$$ is the zero level of a smooth level set function $$\phi$$, i.e., $${\it{\Gamma}}= \{\, x \in {\it{\Omega}} | \phi(x)=0\,\}$$. This level set function is not necessarily close to a distance function, but has the usual properties of a level set function: $$\label{LSdef} \|\nabla \phi(x)\| \sim 1, \,\, \|D^2 \phi(x)\| \leq c \quad \text{for all}\,\, x \,\, \text{in a neighborhood U of {\it{\Gamma}}}.$$ (2.1) We assume that the level set function has the smoothness property $$\phi \in C^{k+2}(U)$$. The assumptions on the level set function (2.1) imply the following relation, which is fundamental in the analysis below $$\label{eq1} |\phi(x+\epsilon \nabla \phi(x)) - \phi(x+\tilde \epsilon \nabla \phi(x))| \sim |\epsilon - \tilde \epsilon| \quad x \in U,$$ (2.2) for $$|\epsilon|, |\tilde \epsilon|$$ sufficiently small. As an input for the parametric mapping, we need an approximation $$\phi_h \in V_h^k$$ of $$\phi$$, and we assume that this approximation satisfies the error estimate $$\label{err2} \max_{T\in {\mathcal T}} |\phi_h - \phi|_{m,\infty,T \cap U} \lesssim h^{k+1-m},\quad 0 \leq m \leq k+1.$$ (2.3) Here $$|\cdot|_{m,\infty,T\cap U}$$ denotes the usual seminorm on the Sobolev space $$H^{m,\infty}(T\cap U)$$. Note that (2.3) holds for the nodal interpolation $$\phi_h = I_k \phi$$ and implies the estimate $$\label{err1} \|\phi_h - \phi\|_{\infty, U} + h\|\nabla(\phi_h - \phi)\|_{\infty, U} \lesssim h^{k+1}.$$ (2.4) Here, and in the remainder, we use the notation $$\lesssim$$ (and $$\sim$$ for $$\lesssim$$ and $$\gtrsim$$), which denotes an inequality with a constant that is independent of $$h$$ and of how the interface $${\it{\Gamma}}$$ intersects the triangulation $${\mathcal T}$$. This constant may depend on $$\phi$$ and on the diffusion coefficient $$\alpha$$, cf. (1.1a). In particular, the estimates that we derive are not uniform in the jump in the diffusion coefficient $$\alpha_1/\alpha_2$$. The zero level of the finite element function $$\phi_h$$ (implicitly) characterizes the discrete interface. The piecewise linear nodal interpolation of $$\phi_h$$ is denoted by $${\hat \phi_h} = I_1 \phi_h$$. Hence, $${\hat \phi_h}(x_i)=\phi_h(x_i)$$ at all vertices $$x_i$$ in the triangulation $${\mathcal T}$$. The low-order geometry approximation of the interface, which is needed in our discretization method, is the zero level of this function, $${{\it{\Gamma}}^{\mathrm{lin}}} := \{{\hat \phi_h} = 0\}$$. The corresponding subdomains are denoted as $${{\it{\Omega}}^{\mathrm{lin}}_i} = \{{\hat \phi_h} \lessgtr 0 \}$$. All elements in the triangulation $${\mathcal T}$$ which are cut by $${{\it{\Gamma}}^{\mathrm{lin}}}$$ are collected in the set $${{\mathcal T}^{{\it{\Gamma}}}} := \{T \in {\mathcal T}, T \cap {{\it{\Gamma}}^{\mathrm{lin}}} \neq \emptyset \}$$. The corresponding domain is $${{\it{\Omega}}^{{\it{\Gamma}}}} := \{x \in T, T\in {{\mathcal T}^{{\it{\Gamma}}}}\}$$. The extended set, which includes all neighbors that share at least one vertex with elements in $${{\mathcal T}^{{\it{\Gamma}}}}$$, is $${{\mathcal T}^{{\it{\Gamma}}}_+} := \{T \in {\mathcal T}, T\cap {{\it{\Omega}}^{{\it{\Gamma}}}} \neq \emptyset \}$$ with the corresponding domain $${{\it{\Omega}}^{{\it{\Gamma}}}_+} := \{x \in T, T \in {{\mathcal T}^{{\it{\Gamma}}}}\}$$. 2.2 Projection operator onto the finite element space $$V_h^k({\it{\Omega}}^{\it{\Gamma}})^d$$ In the construction of the isoparametric mapping $${\it{\Theta}}_h$$, we need a projection step from a function which is piecewise polynomial, but discontinuous (across element interfaces) to the space of continuous finite element functions. Let $$C({{\mathcal T}^{{\it{\Gamma}}}}) = \bigoplus\limits_{T\in {{\mathcal T}^{{\it{\Gamma}}}}} C(T)$$ and $$V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}}) := V_h^k|_{{{\it{\Omega}}^{{\it{\Gamma}}}}}$$. We introduce a projection operator $$P_h^{\it{\Gamma}}: C({{\mathcal T}^{{\it{\Gamma}}}})^d \rightarrow V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$. The projection operator relies on a nodal representation of the finite element space $$V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})$$. The set of finite element nodes $$x_i$$ in $${{\mathcal T}^{{\it{\Gamma}}}}$$ is denoted by $$N({{\mathcal T}^{{\it{\Gamma}}}})$$, and $$N(T)$$ denotes the set of finite element nodes associated with $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. All elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$ that contain the same finite element node $$x_i$$ form the set denoted by $$\omega(x_i)$$: \begin{align*} \omega(x_i) & := \{T \in {{\mathcal T}^{{\it{\Gamma}}}} \,{|}\, x_i \in N(T) \}, x_i \in N({{\mathcal T}^{{\it{\Gamma}}}}). \end{align*} For each finite element node, we define the local average as $$\label{average} A_{x_i}(v):= \frac{1}{|\omega(x_i)|} \sum_{T \in \omega(x_i)} v_{|T}(x_i), x_i \in N({{\mathcal T}^{{\it{\Gamma}}}}),$$ (2.5) where $$|\cdot|$$ denotes the cardinality of the set $$\omega(x_i)$$. The projection operator $$P_h^{\it{\Gamma}}: C({{\mathcal T}^{{\it{\Gamma}}}})^d \to V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$ is defined as \begin{equation*} P_h^{\it{\Gamma}} v:= \sum_{x_i \in N({{\mathcal T}^{{\it{\Gamma}}}})} A_{x_i}(v) \, \psi_i, \quad v \in C({{\mathcal T}^{{\it{\Gamma}}}})^d, \end{equation*} where $$\psi_i$$ is the nodal basis function corresponding to $$x_i$$. This is a simple and well-known projection operator considered also in, e.g., Oswald (1993, Eqs. (25)–(26)) and Guermond & Ern (2015). Note that $$P_h^{\it{\Gamma}} w = I_k w \,\,\forall \,\, w \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d, \quad \|P_h^{\it{\Gamma}} w\|_{\infty,{\it{\Omega}}} \lesssim \max_{T \in{{\mathcal T}^{{\it{\Gamma}}}}} \| w\|_{\infty,T} \,\, \forall \,\, w \in C({{\mathcal T}^{{\it{\Gamma}}}})^d.$$ (2.6) 3. The isoparametric mapping $$\boldsymbol{{\it{\Theta}}}_{\boldsymbol{h}}$$ In this section, we introduce the transformation $${\it{\Theta}}_h \in (V_h^k)^d$$, which is a bijection on $${\it{\Omega}}$$ and satisfies $${\it{\Theta}}_h = {\rm id}$$ on $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$. This mapping is constructed in two steps. First, a local mapping $${\it{\Theta}}_h^{\it{\Gamma}}$$, which is defined on $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, is derived, and this local mapping is then extended to the whole domain. We also need another bijective mapping on $${\it{\Omega}}$$, denoted by $${\it{\Psi}}$$, which is constructed in a similar two-step procedure. This mapping $${\it{\Psi}}$$ is needed in the error analysis and in the derivation of important properties of $${\it{\Theta}}_h$$. In the higher-order FEM, which is presented in Section 4, the mapping $${\it{\Theta}}_h$$ is a key component, and the mapping $${\it{\Psi}}$$ is not used. In this section, we introduce both $${\it{\Theta}}_h$$ and $${\it{\Psi}}$$, because the construction of both mappings has strong similarities. Since the construction is not standard and consists of a two-step procedure, we outline our approach, cf. Fig. 2: We start with the relatively simple definition of the local mapping $${\it{\Psi}}^{\it{\Gamma}}$$, which has the property $${\it{\Psi}}^{\it{\Gamma}}({{\it{\Gamma}}^{\mathrm{lin}}})={\it{\Gamma}}$$ (Section 3.1), but is only defined in $${\it{\Omega}}^{\it{\Gamma}}$$. In general, this mapping cannot (efficiently) be constructed in practice. The definition of $${\it{\Psi}}^{\it{\Gamma}}$$ is slightly modified, to allow for a computationally efficient construction, which then results in the local isoparametric mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ with the property $${{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})={\it{\Gamma}}_h \approx {\it{\Gamma}}$$ (Section 3.2). $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is only defined in $${\it{\Omega}}^{\it{\Gamma}}$$. The extension of the mappings $${\it{\Psi}}^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is based on the same general procedure, which is derived from extension techniques that are standard in the literature on isoparametric FEMs (cf. Lenoir, 1986; Bernardi, 1989). This general procedure is explained in Section 3.3. The general extension procedure is applied to the mapping $${\it{\Psi}}^{\it{\Gamma}}$$, resulting in the global bijection $${\it{\Psi}}$$. Important properties of $${\it{\Psi}}$$ are derived (Section 3.4). Finally, the general extension procedure is applied to the mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, resulting in the global bijection $${{\it{\Theta}}_h}$$. Important properties of $${{\it{\Theta}}_h}$$ are derived (Section 3.5). Fig. 2. View largeDownload slide Sketch of different mesh transformations $${\it{\Psi}}^{\it{\Gamma}}$$, $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, $${\it{\Psi}}$$, $${\it{\Theta}}_h$$ and $${\it{\Phi}}_h := {\it{\Psi}} \circ {\it{\Theta}}_h^{-1}$$. Fig. 2. View largeDownload slide Sketch of different mesh transformations $${\it{\Psi}}^{\it{\Gamma}}$$, $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, $${\it{\Psi}}$$, $${\it{\Theta}}_h$$ and $${\it{\Phi}}_h := {\it{\Psi}} \circ {\it{\Theta}}_h^{-1}$$. 3.1 Construction of $${\it{\Psi}}^{\it{\Gamma}}$$ We introduce the search direction $$G:=\nabla \phi$$ and a function $$d: {{\it{\Omega}}^{{\it{\Gamma}}}} \to {\mathbb{R}}$$ defined as follows: $$d(x)$$ is the (in absolute value) smallest number such that $$\label{cond1} \phi(x + d(x) G(x))=\hat \phi_h(x) \quad \text{for} \,\, x \,\, \in {{\it{\Omega}}^{{\it{\Gamma}}}}.$$ (3.1) (Recall that $$\hat\phi_h$$ is the piecewise linear nodal interpolation of $$\phi_h$$.) We summarize a few properties of the function $$d$$. Lemma 3.1 For $$h$$ sufficiently small, the relation (3.1) defines a unique $$d(x)$$ and $$d \in C({{\it{\Omega}}^{{\it{\Gamma}}}}) \cap H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}}) \cap C^{k+1}({{\mathcal T}^{{\it{\Gamma}}}})$$. Furthermore, there holds \begin{align} |d(x_i)| & \lesssim h^{k+1} \quad \text{for all vertices $x_i$ of} \,\, T \in {{\mathcal T}^{{\it{\Gamma}}}},\label{resd4a0}\\ \end{align} (3.2a) \begin{align} \|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} &\lesssim h^2, \label{resd4a} \\ \end{align} (3.2b) \begin{align} \|d\|_{H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})} &\lesssim h, \label{resd4b} \\ \end{align} (3.2c) \begin{align} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d\|_{H^{l,\infty}(T)} &\lesssim 1, \,\, \text{for} \,\, l \leq k+1. \label{resd4c} \end{align} (3.2d) Proof. For $$|\alpha| \leq \alpha_0 h$$, with a fixed $$\alpha_0 >0$$, we introduce for a fixed $$x \in {{\it{\Omega}}^{{\it{\Gamma}}}}$$ the continuous function $g(\alpha):=\phi(x+ \alpha G(x))- \hat \phi_h(x).$ From (2.4), one obtains $$\|\hat \phi_h - \phi\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^2$$. Using this, a triangle inequality and (2.1), we get $$\label{lpp} g(\alpha)= \alpha \|\nabla \phi(x)\|_2^2 + \mathcal{O}(h^2).$$ (3.3) Hence, there exists $$h_0 >0$$ such that for all $$0< h\leq h_0$$ and all $$x\in {{\it{\Omega}}^{{\it{\Gamma}}}}$$, the equation $$g(\alpha)=0$$ has a unique solution $$\alpha =:d(x)$$ in $$[-\alpha_0 h,\alpha_0 h]$$. The result in (3.2b) follows from (3.3). Using (2.2) and (2.4) we get, for a vertex $$x_i$$ of $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, $|d(x_i)| \sim |\phi(x_i+d(x_i)G(x_i))-\phi(x_i)|= |\hat\phi_h(x_i)-\phi(x_i)|=|\phi_h(x_i)-\phi(x_i)| \lesssim h^{k+1},$ which is the estimate in (3.2a). The continuity of $$d$$ on $${{\it{\Omega}}^{{\it{\Gamma}}}}$$ follows from the continuity of $$\phi, G$$ and $$\hat \phi$$. For (3.2c), we differentiate (3.1) and write $$y = x + d(x) G(x)$$: \begin{align*} \nabla \phi(y) + \nabla d \nabla \phi(y)^T G + d (D G^T) \nabla \phi(y) & = \nabla {\hat \phi_h}(x). \end{align*} This yields \begin{align*} \nabla d (\Vert \nabla \phi \Vert_2^2 + \mathcal{O}(h^2)) + \mathcal{O}(h^2) & = \nabla {\hat \phi_h}(x) - \nabla \phi(y) = \nabla {\hat \phi_h}(x) - \nabla \phi(x) + \mathcal{O}(h^2), \end{align*} which, in combination with $$\|\nabla {\hat \phi_h} - \nabla \phi\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h$$, implies $$\Vert \nabla d(x) \Vert \lesssim h$$ a.e. in $${{\it{\Omega}}^{{\it{\Gamma}}}}$$. For $$T\in {{\mathcal T}^{{\it{\Gamma}}}}$$, we consider the function $$F(x,y) = \phi(x+y G(x))-{\hat \phi_h}(x), (x,y)\in T \times (-\alpha_0 h,\alpha_0 h)$$. The function $$d(x)=y(x)$$ solves the implicit equation $$F(x,y(x))=0$$ on $$T$$. We recall the regularity assumption $$\phi\in C^{k+2}(U)$$. From the implicit function theorem, we know that $$y \in C^{k+1}(T)$$, because $$F \in C^{k+1}\big(T \times (- \alpha_0 h, \alpha_0 h)\big)$$. From $$y \in C^{k+1}({{\mathcal T}^{{\it{\Gamma}}}})$$ and $$y \in C({{\it{\Omega}}^{{\it{\Gamma}}}})$$, we conclude $$y \in H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})$$. Note that $$D^\alpha \hat \phi_h=0$$ for $$|\alpha| \geq 2$$ and $$\|D^\alpha \hat \phi_h\|_{\infty,T} \lesssim \| \phi \|_{H^{2,\infty}(T)}$$ for $$|\alpha| \leq 1$$. Hence, $$\label{lp} \|D^\alpha_{(x,y)}F\|_{\infty, T \times (\alpha_0h, \alpha_0 h)} \lesssim \|\phi\|_{H^{l+1,\infty}(U)}\quad \text{for} \,\, |\alpha| \leq k+1.$$ (3.4) Differentiating $$F(x,y(x))=0$$ yields $$\label{eq:impl:1} D^\alpha y(x) = -D_y F(x,y(x))^{-1} D_x^{\alpha} F(x,y(x)) = -A(x) D_x^{\alpha} F(x,y(x)), \quad |\alpha|=1,$$ (3.5) with $$A(x) = S(x)^{-1}$$, $$S(x) = D_y F(x,y(x))= \nabla \phi(x+yG(x))^T\nabla \phi(x) \in [c_0,c_1]$$ with $$c_0, c_1 > 0$$ independent of $$h,x,T$$. Differentiating $$S(x) A(x) = I$$ we get $$\label{eq:impl:2} D^\alpha A(x) = - A(x)^2 D^\alpha S(x), \quad |\alpha|=1.$$ (3.6) From the relations (3.5) and (3.6), it follows that $$|D^\alpha y(x)|$$, $$|\alpha| = l$$ can be bounded in terms of $$\|A(x)\|$$ and $$|D^\alpha_{(x,y)} F(x,y(x))|$$, $$|\alpha|\leq l$$. Combining this with (3.4) proves (3.2d). □ Given the function $$dG \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d \cap H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$, we define $$\label{psi1} {{\it{\Psi}}^{\it{\Gamma}}}(x):= x + d(x) G(x), \quad x \in {{\it{\Omega}}^{{\it{\Gamma}}}}.$$ (3.7) Note that the function $$d$$ and mapping $${\it{\Psi}}^{\it{\Gamma}}$$ depend on $$h$$, through $$\hat \phi_h$$ in (3.1). We do not show this dependence in our notation. As a direct consequence of the estimates derived in Lemma 3.1 and the smoothness assumption $$G=\nabla \phi \in C^{k+1}(U)$$, we have the following uniform bounds on (higher) derivatives of $${\it{\Psi}}^{\it{\Gamma}}$$. Corollary 3.2 The following holds: \begin{align} \|{{\it{\Psi}}^{\it{\Gamma}}} - {\rm id} \|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} +h \|D{{\it{\Psi}}^{\it{\Gamma}}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} & \lesssim h^2, \label{fistderPsi}\\ \end{align} (3.8) \begin{align} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \Vert D^l {{\it{\Psi}}^{\it{\Gamma}}} \Vert_{\infty,T} & \lesssim 1, \quad l \leq k+1.\label{fistderPsiA} \end{align} (3.9) 3.2 Construction of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ The construction of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ consists of two steps. In the first step, we introduce a discrete analogon of $${\it{\Psi}}^{\it{\Gamma}}$$ defined in (3.7), denoted by $${\it{\Psi}}_h^{\it{\Gamma}}$$. Based on this $${\it{\Psi}}_h^{\it{\Gamma}}$$, which can be discontinuous across element interfaces, we obtain a continuous transformation $${{\it{\Theta}}_h^{\it{\Gamma}}} \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$ by averaging with the projection operator $$P_h^{\it{\Gamma}}$$ from Section 2.2. For the construction of $${\it{\Psi}}_h^{\it{\Gamma}}$$, we need an efficiently computable good approximation of $$G=\nabla \phi$$ on $${{\mathcal T}^{{\it{\Gamma}}}}$$. For this, we consider the following two options $$\label{eq:gh} G_h(x) = \nabla \phi_h(x) \quad \text{or} \,\, G_h(x) = (P_h^{\it{\Gamma}} \nabla \phi_h)(x), \quad x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (3.10) Lemma 3.3 Both for $$G_h = P_h^{\it{\Gamma}} \nabla \phi_h$$ and $$G_h= \nabla \phi_h$$, the estimate $$\label{eqlem2} \|G_h - G\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + h \max_{T\in {{\mathcal T}^{{\it{\Gamma}}}}} \| D (G_h - G) \|_{\infty, T} \lesssim h^{k}$$ (3.11) holds with $$G = \nabla \phi$$. Proof. For $$G_h= \nabla \phi_h$$, the result is a direct consequence of (2.4). For $$G_h = P_h^{\it{\Gamma}} \nabla \phi_h$$, we use \begin{align*} \Vert P_h^{\it{\Gamma}} \nabla \phi_h - \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \leq \Vert P_h^{\it{\Gamma}} \nabla \phi_h - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + \Vert \nabla \phi - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}}, \\ \Vert P_h^{\it{\Gamma}} \nabla \phi_h - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} = \Vert P_h^{\it{\Gamma}} (\nabla \phi_h - \nabla \phi) \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim \max_{T\in{{\mathcal T}^{{\it{\Gamma}}}}} \Vert \nabla \phi_h - \nabla \phi \Vert_{\infty,T}, \end{align*} and (2.4) to estimate $$\|G_h - G\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^k$$. For the derivative, we make use of (2.3) and (2.4): \begin{align*} \Vert D (G_h-G) \Vert_{\infty,T} & \leq \Vert D(G_h - I_k G) \Vert_{\infty,T} + \Vert D(G - I_k G) \Vert_{\infty,T} \\ & \lesssim \frac{1}{h} \Vert G_h - I_k G \Vert_{\infty,T} + h^{k-1} \\ & \lesssim \frac{1}{h} \Vert G_h - G \Vert_{\infty,T} + \frac{1}{h} \Vert G - I_k G \Vert_{\infty,T} + h^{k-1} \lesssim h^{k-1}, \end{align*} with estimates that are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. □ Remark 3.4 We comment on the options in (3.10). An important property of the search direction $$G_h$$ is the proximity to a continuous vector field $$G$$, which describes the direction with respect to which the interface $${\it{\Gamma}}$$ can be interpreted as a graph on $${{\it{\Gamma}}^{\mathrm{lin}}}$$. For each $$x \in {\it{\Gamma}}$$, there exists a unique $$y \in {{\it{\Gamma}}^{\mathrm{lin}}}$$ and $$d\in {\mathbb{R}}$$, s.t. $$x = y + d\, G(x)$$. The choices in (3.10) are accurate approximations of $$G(x) = \nabla \phi$$, which (locally) have the graph property for sufficiently smooth interfaces $${\it{\Gamma}}$$. Despite the discontinuities across element interfaces, the discrete search direction $$G_h = \nabla \phi_h$$ is a reasonable choice, as the distance to the continuous search direction $$G=\nabla \phi$$ is sufficiently small, cf. Lemma 3.3. Let $$\mathcal{E}_T \phi_h$$ be the polynomial extension of $$\phi_h|_T$$. We define a function $$d_h: {{\mathcal T}^{{\it{\Gamma}}}} \to [-\delta,\delta]$$, with $$\delta > 0$$ sufficiently small, as follows: $$d_h(x)$$ is the (in absolute value) smallest number such that $$\label{eq:psihmap} \mathcal{E}_T \phi_h(x + d_h(x) G_h(x)) = \hat \phi_h, \quad \text{for} \,\, x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (3.12) Clearly, this $$d_h(x)$$ is a ‘reasonable’ approximation of the steplength $$d(x)$$ defined in (3.1). We summarize a few properties of the function $$d_h$$. Lemma 3.5 For $$h$$ sufficiently small, the relation (3.12) defines a unique $$d_h(x)$$ and $$d_h \in C^\infty({{\mathcal T}^{{\it{\Gamma}}}})$$. Furthermore: \begin{gather} d_h(x_i) =0 \quad \text{for all vertices $x_i$ of $T \in {{\mathcal T}^{{\it{\Gamma}}}}$}, \label{resd1} \\ \end{gather} (3.13a) \begin{gather} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d_h\|_{\infty,T} \lesssim h^2, \label{resd4} \\ \end{gather} (3.13b) \begin{gather} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|\nabla d_h\|_{\infty,T} \lesssim 1. \label{resd6} \end{gather} (3.13c) Proof. These results can be derived using arguments very similar to the ones used in the proof of Lemma 3.1. For completeness, such a proof is given in the Appendix A, Section A.1. □ Given the function $$d_h G_h \in C({{\mathcal T}^{{\it{\Gamma}}}})^d$$, we define $$\label{eq:psih} {\it{\Psi}}_h^{\it{\Gamma}}(x) := x + d_h(x) G_h(x) \quad \text{for} \,\, x \in T \in {{\mathcal T}^{{\it{\Gamma}}}},$$ (3.14) which approximates the function $${\it{\Psi}}^{\it{\Gamma}}$$ defined in (3.7). To remove the possible discontinuities of $${\it{\Psi}}_h^{\it{\Gamma}}$$ in $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, we apply the projection to obtain $${{\it{\Theta}}_h^{\it{\Gamma}}}:= P_h^{\it{\Gamma}} {\it{\Psi}}_h^{\it{\Gamma}} = {\rm id} + P_h^{\it{\Gamma}}(d_h G_h).$$ (3.15) Using the results in the Lemmas 3.1 and 3.5, one can derive estimates on the difference $${\it{\Psi}}^{\it{\Gamma}}- {\it{\Psi}}_h^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}- {\it{\Psi}}_h^{\it{\Gamma}}$$. Such results, which are needed in the error analysis, are presented in the next two lemmas. Lemma 3.6 The estimate $$\label{estPsi} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|{\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}\|_{\infty,T} + h \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|D ({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}) \|_{\infty,T} \lesssim h^{k+1}$$ (3.16) holds. Proof. In the Appendix A, Section A.2, we give a proof that is based on the definitions of $${\it{\Psi}}_h^{\it{\Gamma}}$$, $${\it{\Psi}}^{\it{\Gamma}}$$ and the properties of $$d_h$$ and $$d$$ in Lemmas 3.5 and 3.1. □ Next, we compare $${\it{\Psi}}^{\it{\Gamma}}$$ with the isoparametric mapping $${\it{\Theta}}_h^{\it{\Gamma}}= P_h {\it{\Psi}}_h^{\it{\Gamma}}$$. Lemma 3.7 The estimate $$\sum_{r=0}^{k+1} h^r \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \lesssim h^{k+1}$$ (3.17) holds. Proof. We have with $$P_h v = I_k v$$ for $$v \in C({{\it{\Omega}}^{{\it{\Gamma}}}})$$ \begin{align*} \|{{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} & \leq \|P_h ({\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + \| I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \\ & \lesssim \| {\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}} \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + h^{k+1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\| d G\|_{H^{k+1,\infty}(T)}. \end{align*} With (3.2d) and the smoothness of $$G$$, we have that $$\max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \| d G\|_{H^{k+1,\infty}(T)}$$ is uniformly bounded. Further, we have due to (3.16) $$\| {\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}} \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^{k+1}$$, which proves the estimate for the $$r=0$$ term in the sum. For the $$r=1, \ldots,k+1$$ terms, we note \begin{align*} h^r \Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} & \leq h^r \Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - I_k {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \\ & \lesssim \Vert {{\it{\Theta}}_h^{\it{\Gamma}}} - I_k {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \\ & \lesssim \Vert {{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + \Vert {\it{\Psi}}^{\it{\Gamma}} - I_k {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T}, \end{align*} with estimates that are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. The bound for the first term on the right-hand side follows from the previous result. The other terms are also uniformly bounded by $$h^{k+1}$$ due to the regularity of $${\it{\Psi}}^{\it{\Gamma}}$$, cf. (3.9). □ From the result in Lemma 3.7, we obtain an optimal convergence order result for the distance between the approximate interface $${\it{\Gamma}}_h = {{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})$$ and $${\it{\Gamma}}$$. Lemma 3.8 The estimate ${\rm dist}({\it{\Gamma}}_h,{\it{\Gamma}}) \lesssim h^{k+1}$ holds. Proof. Take $$x \in {\it{\Gamma}}$$. Let $$y=x+\xi \nabla \phi(x) \in {\it{\Gamma}}_h={{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})$$ be the point closest to $$x$$. Using (2.2) and $$\|\nabla \phi(x)\|_2 \sim 1$$, we get, using $$\phi({{\it{\Psi}}^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}}))=0$$, \begin{align*} \|x-y\|_2 & \sim |\xi| \sim |\phi(x+\xi \nabla \phi(x)) - \phi(x)|= |\phi(x+\xi \nabla \phi(x))| \leq \|\phi\|_{\infty,{\it{\Gamma}}_h} \\ & = \|\phi \circ{{\it{\Theta}}_h^{\it{\Gamma}}}\|_{\infty, {{\it{\Gamma}}^{\mathrm{lin}}}} = \|\phi\circ{{\it{\Theta}}_h^{\it{\Gamma}}}- \phi\circ{{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty, {{\it{\Gamma}}^{\mathrm{lin}}}} \lesssim \|{{\it{\Theta}}_h^{\it{\Gamma}}} -{{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^{k+1}. \end{align*} □ 3.3 Extension procedure In this section, we explain and analyse a general extension procedure for extending piecewise smooth functions given on $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ to the domain $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$, in such a way that the extension is zero on $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$ and piecewise smooth on $${{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$. We use a local extension procedure, introduced and analysed in Lenoir (1986); Bernardi (1989), which is a standard tool in isoparametric FEMs. In Sections 3.4 and 3.5, this procedure is applied to construct extensions of $${\it{\Psi}}^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}$$. We first describe an extension operator, introduced in Lenoir (1986), which handles the extension of a function from (only) one edge or face to a triangle or tetrahedron. The presentation here is simpler as in Lenoir (1986), because we restrict to the case $$d=2,3$$ (although this is not essential) and we only treat extension of functions given on the piecewise linear boundary $$\partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$, hence we do not have to consider the issue of boundary parametrizations. Let $$F$$ be an edge or a face of $$T\in {{\mathcal T}^{{\it{\Gamma}}}_+}\setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ from which we want to extend a function $$w \in C(F)$$, which is zero at the vertices of $$F$$, to the interior of $$T$$. $${F}$$ is called a $$1$$-face of $$T$$ if $$F$$ is an edge of the triangle $$T$$ ($$d=2$$) or an edge the tetrahedron $${T}$$ ($$d=3$$), and it is called a $$2$$-face if $$F$$ is a face of the tetrahedron $${T}$$. Applying affine linear transformations $$\hat {\it{\Phi}}_F: \hat{F} \rightarrow F$$ and $$\hat {\it{\Phi}}_T: \hat{T} \rightarrow T$$, we consider the extension problem in the reference configuration, with $$\hat{T}$$ and $$\hat{F}$$ the reference element and a corresponding face (or edge) and $$\hat{w} := w \circ \hat{\it{\Phi}}_F^{-1}$$. By $$\lambda_i, i=1,..,d+1$$, we denote the barycentric coordinates of $$\hat{T}$$, and we assume that the vertices corresponding to the coordinates $$\lambda_1, \ldots, \lambda_{p+1}$$ with coordinates $$a_1,...,a_{p+1} \in {\mathbb{R}}^d$$ are also the vertices of the $$p$$-face $$\hat{F}$$. The linear scalar weight function $$\omega$$ and the vector function $$Z$$, mapping from $$\hat{T}$$ to $$\hat{F}$$, are defined by $$\label{defomega} \omega := \sum_{i=1}^{p+1} \lambda_i \quad \text{and} \quad Z := \left(\sum_{i=1}^{p+1} \lambda_i a_i \right) / \omega.$$ (3.18) Furthermore, given the interpolation operator $${\it{\Lambda}}_l: C(\hat{F}) \rightarrow \mathcal{P}^l(\hat{F})$$, we define $$\label{defA} A_l^\ast := {\rm id} - {\it{\Lambda}}_l, \quad A_l := {\it{\Lambda}}_l - {\it{\Lambda}}_{l-1} = - A_{l}^\ast + A_{l-1}^\ast.$$ (3.19) The interpolation operator $${\it{\Lambda}}_l$$ in Lenoir (1986) is the usual nodal interpolation operator. We note, however, that this choice is not crucial. Given these components, we define the following extension operator from Lenoir (1986); Bernardi (1989): $$\label{eq:lenoirextref} \mathcal{E}^{\hat{F}\rightarrow \hat{T}} \hat w :=\omega^{k+1} A_k^\ast(\hat w)\circ Z+ \sum_{l=2}^k \omega^l A_l(\hat{w}) \circ Z, \quad \hat{w} \in C(\hat{F}).$$ (3.20) The extension in physical coordinates is then given by $$\label{eq:lenoirext} \mathcal{E}^{F \rightarrow T} w := (\mathcal{E}^{\hat{F}\rightarrow \hat{T}} (w \circ \hat{{\it{\Phi}}}_F^{-1})) \circ \hat{{\it{\Phi}}}_T^{-1}.$$ (3.21) A few elementary properties of this extension operator are collected in the following lemma. These results easily follow from Remarks 6.4–6.6 in Bernardi (1989). We use the notation $$C_0^m(F):= \{\, v \in C^m(F) | v|_{\partial F}=0\}$$. We write $$C_0(F):=C_0^0(F)$$. Lemma 3.9 The following holds: \begin{align} (\mathcal{E}^{F \rightarrow T} w)_{|F} & = w_{|F} & \text{for all} \,\, w \in C_0(F),\\ \end{align} (3.22) \begin{align} \mathcal{E}^{F \rightarrow T} w & \in \mathcal{P}^k(T)& \text{for all} \,\, w\in \mathcal{P}^k(F), \label{proppol} \\ \end{align} (3.23) \begin{align} \mathcal{E}^{F \rightarrow T} w & = 0 & \text{for all} \,\, w\in \mathcal{P}^1(F). \label{proppone} \\ \end{align} (3.24) For all $$p$$-faces $$F, \tilde F$$ of $$T$$ with $$\tilde F \neq F$$: \begin{align} (\mathcal{E}^{F \rightarrow T}w)_{|\tilde F} & = 0 & \text{for all $w \in C_0(F)$}. \label{proponeface} \\ \end{align} (3.25) Let $$T,\tilde T$$ be two tetrahedra with a common face $$F_2= T \cap \tilde T$$ and $$F_1$$ an edge of $$F_2$$. Then: \begin{align} (\mathcal{E}^{F_1 \rightarrow T}w)_{|F_2} & = (\mathcal{E}^{F_1 \rightarrow \tilde T}w)_{|F_2} & \text{for all $w \in C_0(F_1)$}. \label{propconsist} \end{align} (3.26) The following result is a key property of this extension operator. Similar results are derived in Lenoir (1986); Bernardi (1989). Lemma 3.10 For any $$p$$-face $$F$$ of $$T$$ the following holds: $$\Vert D^n \mathcal{E}^{F\rightarrow T} w \Vert_{\infty,T} \lesssim \sum_{r=n}^{k+1} h^{r-n} \Vert D^r w \Vert_{\infty,F}, \quad \forall w \in C_0^{k+1}(F), n=0,..,k+1.$$ Proof. The proof is along the same lines as the proof of Theorem 6.2 in Bernardi (1989), and is given in the Appendix A, Section A.3. □ We now describe how to obtain a ‘global’ extension $$w^{ext}$$ of a function $$w \in C(\partial{{\it{\Omega}}^{{\it{\Gamma}}}})$$. The structure of this extension is as follows. First, we introduce a simple variant of the previous extension operator $$\mathcal{E}^{F\rightarrow T}$$ to extend values from vertices and obtain $$w_0^{ext}$$. The difference $$w - w_0^{ext}$$ has, by construction, zero values on all vertices, so that we can apply the extension from edges to elements, using $$\mathcal{E}^{F\rightarrow T}$$ on each element, to obtain $$w_1^{ext}$$. In two dimensions, this already concludes the extension and we set $$w^{ext} = w_1^{ext}$$. In three dimensions we finally apply the extension from faces to elements, based on $$w - w_1^{ext}$$, which has zero values on all edges, and obtain the extended function $$w^{ext}$$. 3.3.1 The extension from vertices We define the linear interpolation $$I_1^{ext}:\, C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}}) \to V_h^1({{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$ as $$\left\{\begin{array}{@{}ll@{}} \displaystyle(I_1^{ext}w)(x_i) = w(x_i) & \text{for all vertices}\,\, x_i \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}} \\ \displaystyle(I_1^{ext}w)(x_i) = 0 & \text{at all other vertices in} \,\,{{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}, \end{array}\right.$$ (3.27) and for $$w \in C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ we set $$w_0^{ext} := I_1^{ext} w$$. With this extension, we have $$(w - w_0^{ext})(x_i) = 0$$ for all vertices $$x_i \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. On all elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ that have no edge in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$, we set $$w^{ext} = w_0^{ext}$$. All other elements are treated below. 3.3.2 The extension from edges Let $$S_1$$ be the set of edges in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal T}(F)$$ the set of elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ which have $$F$$ as an edge. We define the extension from edges as $$\label{eq:extfromedge} w_1^{ext} := w_0^{ext} + \sum_{F \in S_1} \sum_{T\in{\mathcal T}(F)} \mathcal{E}^{F\rightarrow T} (w - w_0^{ext}).$$ (3.28) We note that $$w_1^{ext}$$ is continuous due to the consistency property (3.26). On all elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ that have no face in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$, i.e., all elements in the two-dimensional case, we set $$w^{ext} := w_1^{ext}$$. All other elements are treated below with an additional extension from faces to elements. 3.3.3 The extension from faces Let $$S_2$$ be the set of faces in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal T}(F)$$ the set of elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ which have $$F$$ as a face. Analogously to (3.28), we define the extension from faces as $$w_2^{ext} := w_1^{ext} + \sum_{F \in S_2} \sum_{T\in{\mathcal T}(F)} \mathcal{E}^{F\rightarrow T} \left(w - w_1^{ext}\right)\!.$$ (3.29) With this extension, we finally have $$(w - w_2^{ext})|_{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}} = 0$$ and set $$w^{ext} = w_2^{ext}$$. From the properties of the extension operator listed above, it follows that $$w^{ext} \in C({{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$, $$w^{ext}$$ is a continuous extension of $$w$$ and $$w=0$$ on $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$. This defines the linear extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}:\, C(\partial{{\it{\Omega}}^{{\it{\Gamma}}}}) \to C({{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$, by $$w^{ext}= \mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w$$. This operator is suitable for the extensions of $${{\it{\Theta}}_h^{\it{\Gamma}}} - {\rm id}$$ and $${{\it{\Psi}}^{\it{\Gamma}}}- {\rm id}$$, and inherits the boundedness property given in Lemma 3.10, which leads to the following result. Theorem 3.11 Let $${\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ denote the set of vertices in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ the set of all edges ($$d=2$$) or faces ($$d=3$$) in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. The following estimates hold \begin{align} \left\|D^n\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w\right\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} & \lesssim \max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})} \sum_{r=n}^{k+1} h^{r-n} \|D^rw\|_{\infty,F} \nonumber \\ & \quad +\, h^{-n}\max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|w(x_i)|,\quad n=0,1, \label{estext1} \\ \end{align} (3.30) \begin{align} \max_{T \in {{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \left\|D^n\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w\right\|_{\infty,T} & \lesssim \sum_{r=n}^{k+1} h^{r-n} \|D^rw\|_{\infty,F}, \quad n=2,\ldots, k+1,\label{estext2} \end{align} (3.31) for all $$w \in C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ such that $$w \in C^{k+1}(F)$$ for all $$F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$. Proof. We note that due to (3.24) we have that $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}} = I_1^{ext} + \mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$. Furthermore $$(w - I_1^{ext}w)(x_i)=0$$ for all vertices $$x_i \in \partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$ is a composition of the element-local extension operator $$\mathcal{E}^{F \rightarrow T}$$ only. Hence, the results in Lemma 3.10 imply corresponding results for $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$. For $$r \geq 2$$, we have $$D^rI_1^{ext}w=0$$, and furthermore $$\label{pll} \|D^r(w-I_1^{ext}w)\|_{\infty,F} \lesssim \|D^rw\|_{\infty,F}, \quad \text{for} \,\, 0 \leq r \leq k+1, \,\, \text{and} \,\, F\in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}}).$$ (3.32) From this estimate and the result in Lemma 3.10, the estimate (3.31) follows. Using $\| I_1^{ext}w\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}}+h \|DI_1^{ext}w\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|w(x_i)|$ in combination with the result in Lemma 3.10 and (3.32) yields the estimate (3.30). □ 3.4 The global mesh transformation $${\it{\Psi}}$$ Given the local transformation $${{\it{\Psi}}^{\it{\Gamma}}}$$ and the extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}$$ defined in Section 3.3, we define the following global continuous mapping on $${\it{\Omega}}$$: $$\label{eq:globext} \mathcal{E} {\it{\Psi}}^{\it{\Gamma}} = \left\{\begin{array}{@{}rcl@{}} {{\it{\Psi}}^{\it{\Gamma}}} & \text{on} & {{\it{\Omega}}^{{\it{\Gamma}}}}, \\ {\rm id} +\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id}) & \text{on} & {{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}, \\ {\rm id} & \text{on} & {\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}, \end{array}\right.$$ (3.33) and $${\it{\Psi}}:=\mathcal{E} {\it{\Psi}}^{\it{\Gamma}}$$. For this global mapping, the following bounds on derivatives are easily derived from the results already obtained for $${\it{\Psi}}^{\it{\Gamma}}$$ and for the extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}$$. Theorem 3.12 For $${\it{\Psi}}=\mathcal{E} {\it{\Psi}}^{\it{\Gamma}}$$, the following holds: \begin{align} \|{\it{\Psi}} - {\rm id}\|_{\infty,{\it{\Omega}}}+ h\|D{\it{\Psi}} - I\|_{\infty,{\it{\Omega}}} & \lesssim h^2, \label{bpsi1} \\ \end{align} (3.34) \begin{align} \max_{T\in{\mathcal T}} \Vert D^l {\it{\Psi}} \Vert_{\infty,T} & \lesssim 1, \quad 0 \leq l \leq k+1. \label{bpsi2} \end{align} (3.35) Proof. On $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$, these results are trivial, because $${\it{\Psi}} - I=0$$. On $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, we have $$D{\it{\Psi}} - I= D{{\it{\Psi}}^{\it{\Gamma}}}- I$$, and the results are a direct consequence of Corollary 3.2. On $${{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$, the result in (3.35) for $$l=0,1$$ follows from (3.34), and for $$l \geq 2$$, it follows from (3.31) combined with (3.9). We use the estimate in (3.30) and thus get \begin{align*} & \|{\it{\Psi}} - {\rm id} \|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} + h\|D{\it{\Psi}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ &\quad =\|\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}}+ h \|D\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ &\quad \lesssim \sum_{r=0}^{k+1} h^{r} \max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}\|D^r ({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,F}+ \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})(x_i)| \\ &\quad \lesssim \|{{\it{\Psi}}^{\it{\Gamma}}} - {\rm id}\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}}+h \|D{{\it{\Psi}}^{\it{\Gamma}}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + h^2 \max_{2 \leq l \leq k+1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|D^l {{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty,T} + \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|d(x_i)|, \end{align*} and using (3.2a) and the results of Corollary 3.2 completes the proof. □ From (3.34), it follows that, for $$h$$ sufficiently small, $${\it{\Psi}}$$ is a bijection on $${\it{\Omega}}$$. Furthermore, this mapping induces a family of (curved) finite elements that is regular of order $$k$$, in the sense as defined in Bernardi (1989). The corresponding curved finite element space is given by $$\label{FEcurvedPsi} V_{h,{\it{\Psi}}}:= \{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^k\, \}.$$ (3.36) Due to the results in Theorem 3.12, the analysis of the approximation error for this finite element space as developed in Bernardi (1989) can be applied. Corollary 4.1 from that article yields that there exists an interpolation operator $${\it{\Pi}}_h:\, H^{k+1}({\it{\Omega}}) \to V_{h,{\it{\Psi}}}$$, such that $$\label{errorBernardi} \|u- {\it{\Pi}}_h u\|_{L^2({\it{\Omega}})}+ h\|u-{\it{\Pi}}_h u\|_{H^1({\it{\Omega}})} \lesssim h^{k+1} \|u\|_{H^{k+1}({\it{\Omega}})} \quad \text{for all} u \in H^{k+1}({\it{\Omega}}).$$ (3.37) This interpolation result will be used in the error analysis of our method in Section 5.5. 3.5 The global mesh transformation $${{\it{\Theta}}_h}$$ We define the extension of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ using the same approach as for $${\it{\Psi}}^{\it{\Gamma}}$$ in (3.33), i.e., $${{\it{\Theta}}_h}:=\mathcal{E}{{\it{\Theta}}_h^{\it{\Gamma}}}$$. This global mapping $${{\it{\Theta}}_h}$$ is used in the isoparametric unfitted FEM explained in the next section. Note that $${\it{\Psi}}:=\mathcal{E}{\it{\Psi}}^{\it{\Gamma}}$$ is only used in the analysis. Remark 3.13 The mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is a piecewise polynomial function of degree $$k$$ on $$\partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$ and thus $${{\it{\Theta}}_h} \in V_h^k({\it{\Omega}})^d$$, cf. (3.23). Further, we have $${{\it{\Theta}}_h^{\it{\Gamma}}}(x_i) = 0$$ on all vertices of $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. Both lead to simplifications in the extension of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$. In (3.20), the first term involving $$A_k^{\ast}$$ vanishes as $${{\it{\Theta}}_h^{\it{\Gamma}}}|_F \in \mathcal{P}^k(F)$$ for every edge and face $$F \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. Furthermore, $$I_1^{ext} {{\it{\Theta}}_h^{\it{\Gamma}}} = 0$$, so that in an implementation only the extensions from edges and faces (with $$w_0^{ext} = 0$$) have to be considered. We do not address further implementation aspects of the mapping $${{\it{\Theta}}_h}$$ here. For the local mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, these are discussed in Lehrenfeld (2016a). The extension procedure is the same as in the well-established isoparametric FEM for a high-order boundary approximation. Here, we only note that in our implementation of this extension, we use a convenient approach based on a hierarchical/modal basis for the finite element space $$V_h^k$$ (cf. Karniadakis & Sherwin, 2013). We discuss shape regularity of the mapping $${\it{\Theta}}_h$$. Clearly, the mapping $${\it{\Theta}}_h$$ should be a bijection on $${\it{\Omega}}$$, and the transformed simplices $${\it{\Theta}}_h(T)$$, $$T \in {\mathcal T}$$, should have some shape regularity property. It is convenient to relate the transformed simplices $${\it{\Theta}}_h(T)$$ to (piecewise) transformations of the unit simplex, denoted by $$\hat T$$. The simplicial triangulation $${\mathcal T}$$ can be represented by affine transformations $${\it{\Phi}}_T(x)=A_Tx +b_T$$, i.e., $${\mathcal T}=\{\, {\it{\Phi}}_T(\hat T)\,\}$$. This defines a mapping $$\hat T \to \tilde{T}:={\it{\Theta}}_h ({\it{\Phi}}_T(\hat T))$$. This mapping should be bijective and well conditioned. Note that $$\kappa(D({{\it{\Theta}}_h} \circ {\it{\Phi}}_T)) \leq \kappa(D {{\it{\Theta}}_h}) \kappa(D {\it{\Phi}}_T)$$, where $$\kappa(\cdot)$$ denotes the spectral condition number. Because $${\it{\Phi}}_T: \hat T \to T$$ is bijective and well conditioned, it suffices to show the bijectivity and well-conditioning of $${\it{\Theta}}_h: T \to {\it{\Theta}}_h(T)$$. Note that $${\it{\Psi}}- {{\it{\Theta}}_h}= \mathcal{E}({{\it{\Psi}}^{\it{\Gamma}}}-{{\it{\Theta}}_h^{\it{\Gamma}}})$$ and using the estimate (3.30), with $$n=1$$, combined with the result in Lemma 3.7, we get $$\|D({\it{\Psi}}- {{\it{\Theta}}_h})\|_{\infty,{\it{\Omega}}} \lesssim h$$. Thus, combined with the estimate in (3.34) we get $$\label{estD} \|D{{\it{\Theta}}_h} - I\|_{\infty,{\it{\Omega}}} \lesssim h.$$ (3.38) This implies that for $$h$$ sufficiently small (the ‘resolved’ case) $$D{\it{\Theta}}_h$$ is invertible, and thus $${\it{\Theta}}_h: T \to {\it{\Theta}}_h(T)= \tilde T$$ is a bijection and furthermore $$\kappa(D {\it{\Theta}}_h) = 1+ \mathcal{O}(h)$$. Hence, we have shape regularity of $$\tilde{{\mathcal T}} = \{\tilde T\}$$ for $$h$$ sufficiently small. In the analysis, we consider only the resolved case with $$h$$ sufficiently small. In practice, one needs suitable modifications of the method to guarantee shape regularity also in cases where $$h$$ is not ‘sufficiently small’. We do not discuss this here and instead refer to Lehrenfeld (2016a). Further important consequences of the smallness of the deformation $${{\it{\Theta}}_h}$$ are summarized in the following lemma. Lemma 3.14 Let $$F := D {{\it{\Theta}}_h}$$, $$J_V := \det(F)$$, $$J_{{\it{\Gamma}}} := J_V \Vert F^{-T} n_{{\it{\Gamma}}^{\rm lin}} \Vert_2$$. There holds \begin{align} \max_{T \in {\mathcal T}} \| F - I \|_{\infty,T} & \lesssim h, \label{estF}\\ \end{align} (3.39a) \begin{align} \int_{{\it{\Omega}}_i^{\rm lin}} v^2 \, {\rm d}\tilde x \sim \int_{{\it{\Omega}}_i^{\rm lin}} J_V v^2 \, {\rm d}\tilde x &= \int_{{{\it{\Theta}}_h}({\it{\Omega}}_i^{\rm lin})} \!\!\!\!\!\!\!\! (v \circ {{\it{\Theta}}_h}^{-1})^2 \, {\rm d} x, \label{jacV}\\ \end{align} (3.39b) \begin{align} \label{jacG} \int_{{\it{\Gamma}}^{\rm lin}} v^2 \, {\rm d}\tilde s \sim \int_{{\it{\Gamma}}^{\rm lin}} J_{\it{\Gamma}} v^2 \, {\rm d}\tilde s &= \int_{{{\it{\Theta}}_h}({\it{\Gamma}}^{\rm lin})} \!\!\!\!\!\!\!\! (v \circ {{\it{\Theta}}_h}^{-1})^2 \, {\rm d}s, \\ \end{align} (3.39c) \begin{align} \label{jacgrad} \Vert \nabla (v \circ {{\it{\Theta}}_h}^{-1}) \Vert_2 & \sim \Vert \nabla v \Vert_2 \text{a.e. in} {\it{\Omega}} \text{for} v \in H^1({\it{\Omega}}). \end{align} (3.39d) Proof. From (3.38), the result (3.39a) and $$\det(F)=1+\mathcal{O}(h)$$ easily follow. The latter implies the $$\sim$$ result in (3.39b). We now consider the integral transformation result in (3.39c). We only treat $$d=3$$ ($$d=2$$ is very similar). Take $$x \in {{\it{\Gamma}}^{\mathrm{lin}}} \cap T$$ and a local orthonormal system $$t_1, t_2 \in n_{{{\it{\Gamma}}^{\mathrm{lin}}}}(x)^\perp$$ at $$x$$. The function $$g(z_1,z_2):= {{\it{\Theta}}_h}(x+z_1t_1+z_2 t_2)$$, $$z_i \in \Bbb{R}$$ parametrizes $${{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}} \cap T)$$. For the change in measure, we have, with $$F=F(x)=D{{\it{\Theta}}_h}(x)$$, $$\label{changemeas} \begin{split} {\rm d}s & = \left\| \frac{\partial g}{\partial z_1} \times \frac{\partial g}{\partial z_2}\right\|_2\, {\rm d} \tilde s =\|F t_1 \times F t_2\|_2\, {\rm d}\tilde s \\ & = |\det (F)| \|F^{-T}(t_1 \times t_2)\|_2\, {\rm d}\tilde s = \det (F) \|F^{-T}n_{{{\it{\Gamma}}^{\mathrm{lin}}}}(x)\|_2 \, {\rm d}\tilde s = J_{\it{\Gamma}} \, {\rm d}\tilde s, \end{split}$$ (3.40) which proves the equality in (3.39c). From (3.39a) and $$\det(F)=1+\mathcal{O}(h)$$, we get the $$\sim$$ result in (3.39c). With $$\nabla (v \circ {{\it{\Theta}}_h}^{-1})= F^{-T} \nabla v$$ and (3.39a), the result in (3.39d) follows. □ 4. Isoparametric unfitted FEM In this section, we introduce the isoparametric unfitted FEM based on the isoparametric mapping $${\it{\Theta}}_h$$. We consider the model elliptic interface problem (1.1). The weak formulation of this problem is as follows: determine $$u \in H_0^1({\it{\Omega}})$$ such that $\int_{{\it{\Omega}}} \alpha \nabla u \cdot \nabla v \, {\rm d}x = \int_{\it{\Omega}} f v \, {\rm d}x \quad \text{for all} \,\, v \in H_0^1({\it{\Omega}}).$ We define the isoparametric Nitsche unfitted FEM as a transformed version of the original Nitsche unfitted FE discretization (Hansbo & Hansbo, 2002) with respect to the interface approximation $${\it{\Gamma}}_h = {\it{\Theta}}_h({{\it{\Gamma}}^{\mathrm{lin}}})$$. We introduce some further notation. The standard unfitted space w.r.t. $${{\it{\Gamma}}^{\mathrm{lin}}}$$ is denoted by $V_h^{\it{\Gamma}}:= {V_h^k}_{|{{\it{\Omega}}^{\mathrm{lin}}_1}}\oplus {V_h^k}_{|{{\it{\Omega}}^{\mathrm{lin}}_2}}.$ To simplify the notation, we do not explicitly express the polynomial degree $$k$$ in $$V_h^{\it{\Gamma}}$$. Remark 4.1 Note that the polynomial degree $$k$$ is used in the FE space that contains the level set function approximation, $$\phi_h \in V_h^k$$. In the definition of the unfitted FE space $$V_h^{\it{\Gamma}}$$ above, which is used for the discretization of the interface problem (1.1), we could also use a polynomial degree $$m \neq k$$. We restrict to the case that both spaces (for the discrete level set function and for the discretization of the PDE) use the same degree $$k$$, because this simplifies the presentation and there is no significant improvement of the method if one allows $$m \neq k$$. The isoparametric unfitted FE space is defined as $$\label{transfspace} V_{h,{\it{\Theta}}}^{\it{\Gamma}}:= \{\, v_h \circ {\it{\Theta}}_h^{-1} | v_h \in V_h^{\it{\Gamma}}\, \}= \{\, \tilde v_h | \tilde v_h \circ {\it{\Theta}}_h \in V_h^{\it{\Gamma}}\, \}.$$ (4.1) Based on this space, we formulate a discretization of (1.1) using the Nitsche technique (Hansbo & Hansbo, 2002) with $${\it{\Gamma}}_h = {\it{\Theta}}_h({{\it{\Gamma}}^{\mathrm{lin}}})$$ and $${\it{\Omega}}_{i,h} = {\it{\Theta}}_h({{\it{\Omega}}^{\mathrm{lin}}_i})$$ as numerical approximation of the geometries: determine $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ such that $$\label{Nitsche1} A_h(u_h,v_h) := a_h(u_h,v_h) + N_h(u_h,v_h) = f_h(v_h) \quad \text{for all} v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}},$$ (4.2) with the bilinear forms \begin{align} a_h(u,v) & := \sum_{i=1}^2 \alpha_i \int_{{\it{\Omega}}_{i,h}} \nabla u \cdot \nabla v\, {\rm d}x,\\ \end{align} (4.3a) \begin{align} N_h(u,v) & := N_h^c(u,v) + N_h^c(v,u) + N_h^s(u,v),\\ \end{align} (4.3b) \begin{align} N_h^c(u,v) & := \int_{{\it{\Gamma}}_h} \{\!\!\{{-\alpha \nabla v}\}\!\!\} \cdot n [\hspace{-2pt}[{u}]\hspace{-2pt}]\, {\rm d}s, \quad N_h^s(u,v) := \bar \alpha \frac{\lambda}{h} \int_{{\it{\Gamma}}_h} [\hspace{-2pt}[{u}]\hspace{-2pt}] [\hspace{-2pt}[{v}]\hspace{-2pt}]\, {\rm d}s \end{align} (4.3c) for $$u, v \in V_{h,{\it{\Theta}}}^{\it{\Gamma}} + {V_{\text{reg},h}}$$ with $${V_{\text{reg},h}} := H^1({\it{\Omega}}) \cap H^2({\it{\Omega}}_{1,h} \cup {\it{\Omega}}_{1,h})$$. Here, $$n = n_{{\it{\Gamma}}_h}$$ denotes the outer normal of $${\it{\Omega}}_{1,h}$$ and $$\bar \alpha = \frac12(\alpha_1 + \alpha_2)$$ the mean diffusion coefficient. For the averaging operator $$\{\!\!\{{\cdot}\}\!\!\}$$, there are different possibilities. We use $$\{\!\!\{w \}\!\!\} := \kappa_1 w_{|{\it{\Omega}}_{1,h}} + \kappa_2w_{|{\it{\Omega}}_{2,h}}$$ with a ‘Heaviside’ choice where $$\kappa_i = 1$$ if $$|T_i| > \frac12 |T|$$ and $$\kappa_i = 0$$ if $$|T_i| \leq \frac12 |T|$$. Here, $$T_i = T \cap {{\it{\Omega}}^{\mathrm{lin}}_i}$$, i.e., the cut configuration on the undeformed mesh is used. This choice in the averaging renders the scheme in (4.2) stable (for sufficiently large $$\lambda$$) for arbitrary polynomial degrees $$k$$, independent of the cut position of $${\it{\Gamma}}$$, cf. Lemma 5.4 below. A different choice for the averaging which also results in a stable scheme is $$\kappa_i = |T_i|/|T|$$. To define the right-hand side functional $$f_h$$, we first assume that the source term $$f_i :\, {\it{\Omega}}_i \to \Bbb{R}$$ in (1.1a) is (smoothly) extended to $${\it{\Omega}}_{i,h}$$, such that $$f_i= f_{i,h}$$ on $${\it{\Omega}}_i$$ holds. This extension is denoted by $$f_{i,h}$$. We define $$\label{eq:lfs} f_h(v) := \sum_{i=1,2} \int_{{\it{\Omega}}_{i,h}} f_{i,h} v\, {\rm d}x.$$ (4.4) We define $$f_h$$ on $${\it{\Omega}}$$ by $${f_h}_{|{\it{\Omega}}_{i,h}}:= f_{i,h}$$, $$i=1,2$$. For the implementation of this method, in the integrals, we apply a transformation of variables $$y:={\it{\Theta}}_h^{-1}(x)$$. This results in the following representations of the bi- and linear forms: \begin{align} a_h(u,v) & = \sum_{i=1,2} \alpha_i \int_{{{\it{\Omega}}^{\mathrm{lin}}_i}} D{\it{\Theta}}_h^{-T} \nabla u \cdot D{\it{\Theta}}_h^{-T} \nabla v \det (D{\it{\Theta}}_h)\, {\rm d}y, \\ \end{align} (4.5a) \begin{align} f_h(v) & = \sum_{i=1,2} \int_{{{\it{\Omega}}^{\mathrm{lin}}_i}} (f_{i,h} \circ {\it{\Theta}}_h) v \det (D{\it{\Theta}}_h)\, {\rm d}y, \\ \end{align} (4.5b) \begin{align} N_h^c(u,v) & = \int_{{{\it{\Gamma}}^{\mathrm{lin}}}} \det(D {\it{\Theta}}_h) D{\it{\Theta}}_h^{-T} \{\!\!\{{-\alpha \nabla v}\}\!\!\} \cdot (D{\it{\Theta}}_h^{-T} \cdot n^{\text{lin}}) [\hspace{-2pt}[{u}]\hspace{-2pt}]\, {\rm d}y, \label{q} \\ \end{align} (4.5c) \begin{align} N_h^s(u,v) & = \bar \alpha \frac{\lambda}{h} \int_{{{\it{\Gamma}}^{\mathrm{lin}}}} \mathcal{J}_{\it{\Gamma}} [\hspace{-2pt}[{u}]\hspace{-2pt}] [\hspace{-2pt}[{v}]\hspace{-2pt}]\, {\rm d}y, \end{align} (4.5d) where $$\mathcal{J}_{{\it{\Gamma}}} = \det (D{\it{\Theta}}_h) \Vert D {\it{\Theta}}_h^{-T} \cdot n^{\text{lin}} \Vert$$ is the ratio between the measures on $${\it{\Gamma}}_h$$ and $${{\it{\Gamma}}^{\mathrm{lin}}}$$, and $$n^{\text{lin}} = \nabla {\hat \phi_h} / \Vert \nabla {\hat \phi_h} \Vert$$ is the normal to $${{\it{\Gamma}}^{\mathrm{lin}}}$$. We note that in (4.5c) we exploited that the normalization factor $$\Vert D {\it{\Theta}}_h^{-1} \cdot n^{\text{lin}} \Vert$$ for the normal direction $$n$$ of $${\it{\Gamma}}_h$$ cancels out with the corresponding term in $$\mathcal{J}_{{\it{\Gamma}}}$$. Based on this transformation, the implementation of integrals is carried out as for the case of the piecewise planar interface $${{\it{\Gamma}}^{\mathrm{lin}}}$$. The additional variable coefficients $$D {{\it{\Theta}}_h}^{-T}$$, $$\det(D {{\it{\Theta}}_h})$$ are easily and efficiently computable using the property that $${{\it{\Theta}}_h}$$ is a finite element (vector) function. The integrands in (4.5) are in general not polynomial, so that exact integration can typically not be guaranteed. For a discussion of the thereby introduced additional consistency error, we refer to Remark 5.9 in the analysis. 4.1 Numerical experiment In this section, we present results of a numerical experiment for the method (4.2). The domain is $${\it{\Omega}} = [-1.5,1.5]^2$$, and the interface is given as $${\it{\Gamma}} = \{\phi(x) = 0 \}$$ with the level set function $$\phi(x) = \Vert x \Vert_4 - 1$$. Here, $$\Vert x \Vert_4 := (\sum_{i=1}^d x_i^4)^{\frac14}$$. The zero level of $$\phi$$ describes a ‘smoothed square’ and $$\phi$$ is equivalent to a signed distance function in the sense that $$\sqrt[4]{1/2} \leq \Vert \nabla \phi \Vert_2 \leq 1$$, such that $$\mathrm{dist}(x,{\it{\Gamma}}) \leq \sqrt[4]{2} \vert \phi(x) \vert\text{for all}x\in{\it{\Omega}}$$. The level set function $$\phi$$ is approximated with $$\phi_h \in V_h^k$$ by interpolation. For the problem in (1.1), we take the diffusion coefficient $$(\alpha_1,\alpha_2) = (1,2)$$ and the Dirichlet boundary conditions and right-hand side $$f$$, such that the solution is given by $$\label{eq:solution} u(x) = \left\{\begin{array}{@{}rc@{}}\displaystyle 1 + \frac{\pi}{2} - \sqrt{2} \cdot \cos\left(\frac{\pi}{4} \Vert x \Vert_4^4\right), & \ x \in {\it{\Omega}}_1, \\ \displaystyle\frac{\pi}{2} \Vert x \Vert_4\hphantom{)}, & \ x \in {\it{\Omega}}_2. \end{array} \right.$$ (4.6) We note that $$u$$ is continuous on $${\it{\Omega}}$$, but has a kink across the interface $${\it{\Gamma}}$$, such that $$\alpha_1 \nabla u_1 \cdot n_{\it{\Gamma}} = \alpha_2 \nabla u_2 \cdot n_{\it{\Gamma}}$$, cf. the sketch of the solution in Fig. 3. Fig. 3. View largeDownload slide Sketch of domains $${\it{\Omega}}_1$$, $${\it{\Omega}}_2$$ (left), the solution (middle and right) and the initial mesh ($$L=0$$, right) for the example in Section 4.1. Fig. 3. View largeDownload slide Sketch of domains $${\it{\Omega}}_1$$, $${\it{\Omega}}_2$$ (left), the solution (middle and right) and the initial mesh ($$L=0$$, right) for the example in Section 4.1. For the construction of the mapping $${{\it{\Theta}}_h}$$, the one-dimensional nonlinear problem that results from (3.12) is solved with a Newton iteration, cf. (Lehrenfeld, 2016a, section 2.3), up to a (relative) tolerance of $$10^{-14}$$, which required only two to three iterations for each point. For the search direction, we chose $$G_h = \nabla \phi_h$$, cf. (3.10). We also considered $$G_h=P_h^{\it{\Gamma}} \nabla \phi_h$$, which led to very similar results. For the computation of the integrals, we apply the transformation to the reference domains $${{\it{\Omega}}^{\mathrm{lin}}_i}$$ and $${{\it{\Gamma}}^{\mathrm{lin}}}$$ as in (4.5). On each element of the reference domain (e.g., $${{\it{\Omega}}^{\mathrm{lin}}_i} \cap T$$, $$T \in {\mathcal T}_h$$), we apply quadrature rules of exactness degree $$2k-2$$ for the integrals in $$a_h(\cdot,\cdot)$$ and $$N_h^c(\cdot,\cdot)$$, and quadrature rules of exactness degree $$2k$$ for the integrals in $$N_h^s(\cdot,\cdot)$$, see also remark 5.9 below. The method has been implemented in the add-on library ngsxfem to the finite element library NGSolve (Schöberl, 2014). We chose an initial simplicial mesh ($$L=0$$) with $$282$$ elements, which sufficiently resolves the interface such that the shape regularity of the mesh (after transformation) is given without a further limitation step as discussed in Lehrenfeld (2016a, Section 2.6). Starting from this initial triangulation, uniform refinements are applied. The stabilization parameter $$\lambda$$ in the unfitted Nitsche method is chosen as $$\lambda = 20 \cdot k^2$$. With the discrete solution to (4.2) $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, and $$u$$ as in (4.6), we define the following error quantities that we can evaluate for the numerical solutions: \begin{align*} d_{{\it{\Gamma}}_h}\! :=\! \Vert \phi \Vert_{\infty,{\it{\Gamma}}_h}, \quad e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}:= \Vert [\hspace{-2pt}[{E_i u - u_h}]\hspace{-2pt}] \Vert_{L^2({\it{\Gamma}}_h)} \\ e_{L^2}^2\! :=\! \sum_{i=1,2} \Vert E_i u - u_h\Vert_{L^2({\it{\Omega}}_{i,h})}^2, \quad e_{H^1}^2\! :=\! \sum_{i=1,2} \Vert \nabla(E_i u - u_h) \Vert_{L^2({\it{\Omega}}_{i,h})}^2. \end{align*} Here, $$E_i:{\it{\Omega}}_i \rightarrow {\it{\Omega}}_{i,h}={{\it{\Theta}}_h}({{\it{\Omega}}^{\mathrm{lin}}_i})$$ denotes the canonical extension operator for the solution (using the representation in (4.6)). Due to the equivalence of $$\phi$$ to a signed distance function we have $$\mathrm{dist}({\it{\Gamma}}_h,{\it{\Gamma}}) \leq \sqrt[4]{2} d_{{\it{\Gamma}}_h}$$, and thus $$d_{{\it{\Gamma}}_h}$$ yields a (sharp) bound for the error in the geometry approximation. The error quantities $$e_{H^1}$$, $$d_{{\it{\Gamma}}_h}$$ and $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ are also used in the error analysis, cf. (5.1). The results in the experiment confirm the theoretical error bounds. We also include the $$L^2$$-error $$e_{L^2}$$, although in the analysis in this article, we do not derive $$L^2$$-error bounds, cf. Section 6. Results are listed in Table 1. We observe $$d_{{\it{\Gamma}}_h} \sim \mathcal{O}(h^{k+1})$$, $$e_{L^2} \sim \mathcal{O}(h^{k+1})$$, $$e_{H^1} \sim \mathcal{O}(h^{k})$$, i.e., optimal order of convergence in these quantities. The observed convergence rate for the jump across the interface, $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]} \sim \mathcal{O}(h^{k+1})$$, is better than that predicted in the analysis below (by half an order). For a fixed mesh ($$L=const$$), we observe that increasing the polynomial degree $$k$$ dramatically decreases the error in all four error quantities. The discretization with $$k=6$$ on the coarsest mesh $$L=0$$ with only $$6835$$ unknowns (last row in the table) is much more accurate in all the four quantities than the discretization with $$k=1$$ after six additional mesh refinements and 583 401 unknowns (row $$k=1$$, $$L=6$$ in the table). Table 1 Discretization errors and estimated orders of convergence (eoc) for the example inSection 4.1 $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) Table 1 Discretization errors and estimated orders of convergence (eoc) for the example inSection 4.1 $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) Remark 4.2 We note that the arising linear systems depend on the position of the interface within the computational mesh and can become arbitrarily ill-conditioned. To circumvent the impact of this, we used a sparse direct solver for the solution of linear systems. In our experiment, the very poor conditioning of the stiffness matrix for high-order discretizations limited the achievable accuracy to $$e_{L^2} \approx 1\times 10^{-6}$$ and $$e_{H^1} \approx 1\times 10^{-6}$$. Hence, we stopped the refinement if discretization errors in this order of magnitude have been reached. The solution of linear systems is an important issue for high-order unfitted discretizations and requires further attention. To deal with the ill-conditioning, one may consider (new) preconditioning strategies or further stabilization mechanisms in the weak formulation. We leave this as a topic for future research. 5. Error analysis The main new contribution of this article is an error analysis of the method presented in Section 4. We start with the definition of norms and derive properties of the discrete variational formulation in Section 5.1. These results are obtained using the fact that the isoparametric method can be seen as a small perturbation of the standard unfitted Nitsche-XFEM method. To bound the error terms, we use the bijective mapping on $${\it{\Omega}}$$ given by $${\it{\Phi}}_h:= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$. This mapping has the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$ and is close to the identity. Some further relevant properties of this mapping are treated in Section 5.2. Using this mapping, we derive a Strang lemma-type result, which relates the discretization error to consistency errors in the bilinear form and right-hand side functional, and to approximation properties of the finite element space, cf. Section 5.3. In Sections 5.4 and 5.5, we prove bounds for the consistency and approximation error, respectively. Finally, in Section 5.6, an optimal-order $$H^1$$-error bound is given. 5.1 Norms and properties of the bilinear forms In the error analysis, we use the norm \begin{align} \Vert v \Vert_{h}^2 & := \vert v \vert_{1}^2 + \Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2 + \Vert \{\!\!\{{\alpha \nabla v}\}\!\!\} \Vert_{-\frac12,h,{\it{\Gamma}}_h}^2, \label{defn} \\ \end{align} (5.1) \begin{align} \text{with} \Vert v \Vert_{\pm \frac12,h,{\it{\Gamma}}_h}^2 & := \left(\bar{\alpha} / h \right)^{\pm 1} \Vert v \Vert_{L^2({\it{\Gamma}}_h)}^2 \text{and} |v|_1^2:= \sum_{i=1,2} \alpha_i \Vert \nabla v \Vert_{L^2({\it{\Omega}}_{i,h})}^2. \label{ppp} \end{align} (5.2) Note that the norms are formulated with respect to $${\it{\Omega}}_{i,h} = {{\it{\Theta}}_h}({{\it{\Omega}}^{\mathrm{lin}}_i})$$ and $${\it{\Gamma}}_{h} = {{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}})$$, and include a scaling depending on $$\alpha$$. Remark 5.1 For simplicity, we restrict to the setting of a quasi-uniform family of triangulations. In the more general case of a shape-regular (not necessarily quasi-uniform) family of triangulations, one has to replace the interface norm in (5.2) by one with an element wise scaling: $$\label{defr} \Vert v \Vert_{\pm \frac12,h,{\it{\Gamma}}_h}^2 := \sum_{T \in {\mathcal T}^{\it{\Gamma}}} \left(\bar{\alpha} / h_T \right)^{\pm 1} \Vert v \Vert_{L^2({\it{\Gamma}}_{h,T})}^2, \text{with} {\it{\Gamma}}_{h,T} := {{\it{\Theta}}_h}(T\cap {{\it{\Gamma}}^{\mathrm{lin}}}).$$ (5.3) We further define the space of sufficiently smooth functions that allow for the evaluation of normal gradients at the interface $${\it{\Gamma}}_h$$: $$\label{defVreg} {V_{\text{reg},h}} := H^1({\it{\Omega}}) \cap H^2({\it{\Omega}}_{1,h} \cup {\it{\Omega}}_{2,h}).$$ (5.4) Note that the norm $$\|\cdot\|_h$$ and the bilinear forms in (4.3) are well defined on $${V_{\text{reg},h}}+V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. The stability of the method relies on an appropriate choice for the weighting operator $$\{\!\!\{{u}\}\!\!\} = \kappa_1 u_1 + \kappa_2 u_2$$ in (4.2). For the stability analysis of the weighting operator, we introduce the criterion $$\label{eq:crit} \kappa_i^2 \leq c_{\kappa} |T_i|/|T|, i=1,2,$$ (5.5) with a constant $$c_{\kappa}$$ independent of $$\alpha$$, $$h$$ and the cut configuration. We note that this criterion is fulfilled for the Heaviside choice, cf. Section 4, with $$c_{\kappa}=2$$ and the weighting $$\kappa_i =|T_i|/|T|$$ proposed in Hansbo & Hansbo (2002) with $$c_{\kappa}=1$$. Using this criterion, we derive the following inverse estimate, which is an important ingredient in the stability analysis of the unfitted Nitsche method. Lemma 5.2 On a shape-regular (not necessarily quasi-uniform) family of triangulations with an averaging operator $$\{\!\!\{{u}\}\!\!\}$$ satisfying (5.5), there holds $$\label{inverse} \Vert \{\!\!\{{\alpha \nabla u}\}\!\!\} \Vert_{-\frac12,h,{\it{\Gamma}}_h} \lesssim \vert u \vert_{1} \quad \text{for all} \,\, u \in V_h^{\it{\Gamma}}.$$ (5.6) Proof. It suffices to derive the localized estimate for an arbitrary element $$T \in {\mathcal T}^{\it{\Gamma}}$$. Note that for the general case of a not necessarily quasi-uniform family of triangulations, the norm used on the left-hand side in (5.6) is as in (5.3). We only have to show $$h_T \kappa_i^2 \Vert p \Vert_{L^2({\it{\Gamma}}_{h,T})}^2 \lesssim \Vert p \Vert_{L^2({{\it{\Theta}}_h}(T_i))}^2 \quad \text{for all} \,\, p \circ {{\it{\Theta}}_h} \in \mathcal{P}^k(T),$$ with $${\it{\Gamma}}_{h,T} := {{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}_T})$$, $${{\it{\Gamma}}^{\mathrm{lin}}_T} := {{\it{\Gamma}}^{\mathrm{lin}}} \cap T$$ and $$T_i := {{\it{\Omega}}^{\mathrm{lin}}_i} \cap T$$ for $$i=1,2$$. With Lemma 3.14, this is equivalent to the corresponding result on the undeformed domains: $$h_T \kappa_i^2 \Vert p \Vert_{L^2({{\it{\Gamma}}^{\mathrm{lin}}_T})}^2 \lesssim \Vert p \Vert_{L^2(T_i)}^2 \quad \text{for all} \,\, p \in \mathcal{P}^k(T).$$ We transform the problem to the reference element $$\hat{T} := \{x \in {\mathbb{R}}^d | \sum_{i=1}^d x_i \leq 1, x_i \geq 0, i=1,..,d \}$$ with the corresponding affine linear transformation $${\it{\Phi}}_T: \hat{T} \rightarrow T$$, such that $$T = {\it{\Phi}}_T(\hat{T})$$. We introduce $$J_V := {|}\det(D {\it{\Phi}}_T)|$$, $$J_{\it{\Gamma}} := J_V \Vert (D {\it{\Phi}}_T)^{-T} n_{\hat{{\it{\Gamma}}}} \Vert_2$$, $$\hat{{\it{\Gamma}}} = {\it{\Phi}}_T^{-1} ({{\it{\Gamma}}^{\mathrm{lin}}_T})$$ and $$\hat{T}_i := {\it{\Phi}}_T^{-1} (T_i)$$ and then have $$\int_{T_i} p^2 \, {\rm d}x = J_V \int_{\hat{T}_i} (p \circ {\it{\Phi}}_T)^2 \, {\rm d}x$$, and with the same arguments as in (3.40) one gets $$\int_{{{\it{\Gamma}}^{\mathrm{lin}}_T}} p^2 \, {\rm d}s = J_{\it{\Gamma}} \int_{\hat{{\it{\Gamma}}}} (p \circ {\it{\Phi}}_T)^2 \, {\rm d}s.$$ Note that $\|D {\it{\Phi}}_T\|_2^{-1} \leq \Vert (D {\it{\Phi}}_T)^{-T} n_{\hat{{\it{\Gamma}}}} \Vert_2 \leq \|D {\it{\Phi}}_T^{-1}\|_2.$ Due to the assumption of shape regularity, we have $$\|D {\it{\Phi}}_T\|_2 \lesssim h_T$$, $$\|D {\it{\Phi}}_T^{-1}\|_2 \lesssim h_T^{-1}$$ and thus we get $$J_V \sim J_{\it{\Gamma}} h_T$$. Hence, it suffices to show $$\label{investhat} \kappa_i^2 \Vert p \Vert_{\hat{{\it{\Gamma}}}}^2 \leq c_{k,d} \Vert p \Vert_{\hat{T}_i}^2 \quad \text{for all} \,\, p \in \mathcal{P}^k(\hat{T}), i=1,2,$$ (5.7) with a constant $$c_{k,d}$$ depending only on the polynomial degree $$k$$, the dimension $$d$$ and the constant $$c_{\kappa}$$. We now prove the estimate (5.7). As $$\hat{{\it{\Gamma}}}$$ is planar $$\hat{T}_i$$ is a convex polytope. Let $$S$$ be the hyperplane, with normal denoted by $$n_S$$, which contains $$\hat{{\it{\Gamma}}}$$, and $$R = \{x \in S | x + \alpha n_S \in \hat{T}_i, \alpha \in {\mathbb{R}}\}$$ be the normal projection of $$\hat{T}_i$$ into $$S$$, cf. Fig. 4. With $${\mathcal I}_i$$ the subset of those vertices of $$\hat T$$ that are vertices of $$\hat{T}_i$$, we set $$D := \max_{x_V \in {\mathcal I}_i}\mathrm{dist}(S,x_V)$$. We define the cylinder $$T_{R,D} = \{x \in {\mathbb{R}}^d | x = y + \alpha n_S, y \in R, \alpha \in [0,D]\}$$. Note that $$\hat{T}_i \subset T_{R,D}$$ holds. We can bound the volume of $$T_{R,D}$$ by $$|T_{R,D}| \leq |R| D \leq l^{d-1} D$$, where $$l$$ is the maximum distance of two points in $$R$$, which can be bounded by the length of the longest edge of $$\hat{T}$$, i.e., $$l \leq \sqrt{2}$$. Combining this with $$|\hat{T}_i| \geq \frac{\kappa_i^2}{c_{\kappa}} |\hat{T}|$$ from (5.5), we get $$\frac{\kappa_i^2}{c_{\kappa} d!} = \frac{\kappa_i^2}{c_{\kappa}} |\hat{T}| \leq |\hat{T}_i| \leq |T_{R,D}| \leq l^{d-1} D,$$ and thus $$D \geq c_d \frac{\kappa_i^2}{c_{\kappa}}$$ with $$c_d := (d! 2^{\frac{d-1}{2}})^{-1}$$. From this, it follows that there exists a vertex $$x_V$$ of the reference simplex $$\hat T$$ such that $$x_V \in \hat{T}_i$$ and $$\mathrm{dist}(S,x_V) \geq c_d \frac{\kappa_i^2}{c_{\kappa}}$$. Note that $$\mathrm{conv}(\hat{{\it{\Gamma}}},x_V) \subset \hat{T}_i$$. Based on a decomposition of $$\hat{{\it{\Gamma}}}$$ into $$(d-1)$$-simplices $$\{G_l\}_{l=1,..,L}$$, we consider the (nonoverlapping) decomposition of $$\mathrm{conv}(\hat{{\it{\Gamma}}},x_V)$$ into $$d$$-simplices $$T_l:=\mathrm{conv}(G_l,x_V)$$, $$l=1,..,L$$. On $$T_l$$ we apply known inverse trace estimates for polynomials, c.f. (Warburton & Hesthaven, 2003): $$\Vert p \Vert_{L^2(G_l)}^2 \leq \frac{(k+d)(k+1)}{d} \frac{|G_l|}{|T_l|} \Vert p \Vert_{L^2(T_l)}^2$$. With $$|T_l| = \frac{D}{d} |G_l| \geq \frac{\kappa_i^2}{c_{\kappa}} \frac{c_d}{d} |G_l|$$, we obtain $$\kappa_i^2 | G_l | / |T_l| \leq \frac{d}{c_d} c_\kappa$$ and thus, $$\kappa_i^2 \Vert p \Vert_{L^2(G_l)}^2 \leq c_{\kappa} \frac{(k+d)(k+1)}{c_d} \Vert p \Vert_{L^2(T_l)}^2.$$ By summing over $$l=1,..,L$$, this proves (5.7) with $$c_{k,d} = c_{\kappa} d! 2^{\frac{d-1}{2}} (k+d)(k+1)$$. □ Fig. 4. View largeDownload slide Sketch of geometries used in the proof of Lemma 5.2. Fig. 4. View largeDownload slide Sketch of geometries used in the proof of Lemma 5.2. Remark 5.3 The condition (5.5) on the averaging operator is crucial for the estimate (5.6) to hold. Furthermore, the fact that the interface is piecewise planar w.r.t., the reference simplex is used in the analysis. We briefly comment on similar results from the literature. For $$k=1$$ and the weighting $$\kappa_i =|T_i|/|T|$$, the inverse estimate has been shown in Hansbo & Hansbo (2002). In Massjung (2012), the result in Lemma 5.2 has been proven for the Heaviside choice $$\kappa_i \in \{0,1 \}$$ for the case of a higher-order unfitted discontinuous Galerkin discretization for smooth interfaces in two dimensions. A variant of this method has been addressed in Wu & Xiao (2010). We are not aware of any literature in which such a result for higher-order discretizations and dimension $$d \geq 3$$ has been derived. A large part of the analysis of the usual unfitted FEMs can be carried over using the following result: Lemma 5.4 For $$\lambda$$ sufficiently large the estimates \begin{align} A_h(u,u) &\gtrsim \Vert u \Vert_h^2 && \text{for all} \, u \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}, \label{eq:coerc} \\ \end{align} (5.8) \begin{align} A_h(u,v) &\lesssim \Vert u \Vert_h \Vert v \Vert_h && \text{for all} \, u,v \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}} \label{eq:bound} \end{align} (5.9) hold. Proof. We have $$\label{r1} a_h(u,u) \sim |u|_1^2 \quad \text{for} \,\, u \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}}.$$ (5.10) Furthermore, $$\label{r2} \begin{split} |N_h^s(u,v)| & \lesssim \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} \Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}, \quad u,v \in H^1({\it{\Omega}}),\\ N_h^s(u,u) & \sim \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2, \quad u \in H^1({\it{\Omega}}). \end{split}$$ (5.11) For the Nitsche consistency term, we have, for $$u,v \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: $$\label{r3} |N_h^c(u,v)| \lesssim \Vert \{\!\!\{{\alpha \nabla u}\}\!\!\} \Vert_{-\frac12,h,{{\it{\Gamma}}^{\mathrm{lin}}}}\Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{{\it{\Gamma}}^{\mathrm{lin}}}}.$$ (5.12) The result in (5.9) follows from the definition of $$A_h(\cdot,\cdot)$$ and (5.10), (5.11), (5.12). Using (5.6) and the results in (5.10), (5.11), (5.12), we get, for $$u \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, \begin{align*} A_h(u,u) & \gtrsim \vert u \vert_{1}^2 + \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2 - \vert u \vert_{1} \, \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} \\ & \gtrsim \vert u \vert_{1}^2 + \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2, \end{align*} provided $$\lambda$$ is chosen sufficiently large. Using (5.6) again, we obtain the estimate (5.8). □ In the remainder, we assume that $$\lambda$$ is taken sufficiently large, such that the results in Lemma 5.4 hold. Then, the discrete problem (4.2) has a unique solution. 5.2 The bijective mapping $${\it{\Phi}}_h= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$ From the properties derived in Sections 3.4 and 3.5, it follows that, for $$h$$ sufficiently small, the mapping $${\it{\Phi}}_h= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$ is bijection on $${\it{\Omega}}$$ and has the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$. In the remainder, we assume that $$h$$ is sufficiently small such that $${\it{\Phi}}_h$$ is a bijection. It has the smoothness property $${\it{\Phi}}_h \in C({\it{\Omega}})^d \cap C^{k+1}({\mathcal T})^d$$. In the following lemma, we derive further properties of $${\it{\Phi}}_h$$ that will be needed in the error analysis. Lemma 5.5 The following holds: \begin{align} \|{{\it{\Theta}}_h} - {\it{\Psi}}\|_{\infty,{\it{\Omega}}}+ h\|D({{\it{\Theta}}_h} - {\it{\Psi}})\|_{\infty,{\it{\Omega}}} & \lesssim h^{k+1} \label{estt1} \\ \end{align} (5.13) \begin{align} \|{\it{\Phi}}_h - {\rm id}\|_{\infty,{\it{\Omega}}}+ h\|D{\it{\Phi}}_h - I\|_{\infty,{\it{\Omega}}} & \lesssim h^{k+1}.\label{estt2} \end{align} (5.14) Proof. Using the definition $${\it{\Psi}}=\mathcal{E}{{\it{\Psi}}^{\it{\Gamma}}}$$, $${{\it{\Theta}}_h}=\mathcal{E}{{\it{\Theta}}_h^{\it{\Gamma}}}$$ and the results in Theorem 3.11 and Lemma 3.7, we get the result in (5.13) with \begin{align*} &\|{{\it{\Theta}}_h} - {\it{\Psi}}\|_{\infty,{\it{\Omega}}} + h\|D({{\it{\Theta}}_h} - {\it{\Psi}})\|_{\infty,{\it{\Omega}}} \\ &\quad \lesssim \|{{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + h \|D({{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ & \qquad +\max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})} \sum_{r=0}^{k+1} h^{r} \|D^r ({{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}})\|_{\infty,F} +\max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|({{\it{\Theta}}_h^{\it{\Gamma}}}-{{\it{\Psi}}^{\it{\Gamma}}})(x_i)| \\ &\quad \lesssim \sum_{r=0}^{k+1} h^r \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \lesssim h^{k+1}. \end{align*} Note that $${\it{\Phi}}_h-{\rm id}=({\it{\Psi}}-{{\it{\Theta}}_h}){{\it{\Theta}}_h}^{-1}$$. From the results in (3.34) and (5.13), it follows that $$\|{{\it{\Theta}}_h}^{-1}\|_{\infty,{\it{\Omega}}} \lesssim 1$$, $$\|D{{\it{\Theta}}_h}^{-1}\|_{\infty,{\it{\Omega}}} \lesssim 1$$. Hence, the estimate in (5.14) follows from the one in (5.13). □ Remark 5.6 With similar arguments as used in the proof above, one can also derive the bound $\max_{T \in {\mathcal T}} \Vert D^l ({\it{\Phi}}_h - {\rm id}) \Vert_{\infty,T} = \max_{T \in {\mathcal T}} \Vert D^l {\it{\Phi}}_h \Vert_{\infty,T} \lesssim h^{k+1-l}, \qquad T \in {\mathcal T}, l=2,..,k+1,$ but we do not need this in the error analysis. 5.3 Strang lemma In this section, we derive a Strang lemma in which the discretization error is related to approximation and geometry errors. We use the homeomorphism $${\it{\Phi}}_h: {\it{\Omega}} \to {\it{\Omega}}$$ with the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$. We define $${V_{\text{reg}}} :=H^1({\it{\Omega}})\cap H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$, cf. (5.4). Related to $${\it{\Phi}}_h$$ we define $V_{h,{\it{\Phi}}}^{\it{\Gamma}} := \{v \circ {\it{\Phi}}_h^{-1}, v \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}\} \subset H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2),$ and the linear and bilinear forms $$A(u,v) := a(u,v) + N(u,v), \quad f(v) := \sum_{i=1,2} \int_{{\it{\Omega}}_i} f_i v \, {\rm d}x, \quad u,v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}},$$ (5.15) with the bilinear forms $$a(\cdot,\cdot)$$, $$N(\cdot,\cdot)$$ as in (4.3) with $${\it{\Omega}}_{i,h}$$ and $${\it{\Gamma}}_h$$ replaced with $${\it{\Omega}}_i$$ and $${\it{\Gamma}}$$, respectively. In the following lemma, a consistency result is given. Lemma 5.7 Let $$u\in {V_{\text{reg}}}$$ be a solution of (1.1). The following holds: $$\label{eq:cons1} A(u,v) = f(v) \quad \text{for all} \,\, v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}}.$$ (5.16) Proof. Take $$v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}}$$. Since $$[\hspace{-2pt}[{u}]\hspace{-2pt}]_{|{\it{\Gamma}}}=0$$ a.e. on $${\it{\Gamma}}$$, we get $$N^s(u, v)= N^c(v,u)=0$$. Thus, we obtain \begin{align*} A(u,v)- f(v) & = a(u,v)+ N^c(u, v) - \sum_{i=1,2} \int_{{\it{\Omega}}_i} f_i v \, {\rm d}x \\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \alpha_i \nabla u \cdot \nabla v - f_i v \, {\rm d}x +\int_{{\it{\Gamma}}} \{\!\!\{{-\alpha \nabla u \cdot n}\}\!\!\} [\hspace{-2pt}[{v}]\hspace{-2pt}] \, {\rm d}s\\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \alpha_i \nabla u \cdot \nabla v - f v_i \, {\rm d}x+ \int_{{\it{\Gamma}}} [\hspace{-2pt}[{-\alpha \nabla u \cdot n v}]\hspace{-2pt}] \, {\rm d}s, \end{align*} where in the last equality we used the flux continuity in (1.1b). Applying partial integration on $${\it{\Omega}}_i$$ and using (1.1a) results in $A(u, v)- f(v)=\sum_{i=1}^2 \int_{{\it{\Omega}}_i}- {\textrm{div}\ \!} (\alpha_i \nabla u) v - f_i v \, {\rm d}x =0,$ which completes the proof. □ Lemma 5.8 Let $$u\in {V_{\text{reg}}}$$ be a solution of (1.1) and $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ the solution of (4.2). The following holds: $$\label{eq:strang} \begin{split} \Vert u \circ {\it{\Phi}}_h - u_h \Vert_h \lesssim & \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert u\circ {\it{\Phi}}_h - v_h \Vert_h \\ & + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| f_h(w_h) - f(w_h \circ {\it{\Phi}}_h^{-1}) |}{\Vert w_h \Vert_h} \\ & + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| A_h(u\circ {\it{\Phi}}_h,w_h) - A(u,w_h \circ {\it{\Phi}}_h^{-1}) |}{\Vert w_h \Vert_h}. \end{split}$$ (5.17) Proof. The proof is along the same lines as in the well-known Strang Lemma. We use the notation $$\tilde u =u \circ {\it{\Phi}}_h$$ and start with the triangle inequality, where we use an arbitrary $$v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: \begin{equation*} \Vert \tilde u - u_h \Vert_h \leq \Vert \tilde u - v_h \Vert_h + \Vert v_h - u_h \Vert_h. \end{equation*} With $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$-coercivity, cf. (5.8), we have $$\label{eq:strang2} \begin{split} \Vert u_h - v_h \Vert_h^2 & \lesssim A_h(u_h - v_h,u_h - v_h) \stackrel{w_h=u_h-v_h}{=} A_h(u_h-v_h,w_h) \\ & = \left| A_h(\tilde u-v_h,w_h) \right| + \left| A_h(\tilde u,w_h) - f_h(w_h) \right|\!. \end{split}$$ (5.18) Using continuity (5.9) and dividing by $$\Vert w_h \Vert_h = \Vert u_h - v_h \Vert_h$$ results in \begin{equation*} \Vert \tilde u - u_h \Vert_h \lesssim \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert \tilde u - v_h \Vert_h + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| A_h(\tilde u,w_h) - f_h(w_h) |}{\Vert w_h \Vert_h}. \end{equation*} Using the consistency property of Lemma 5.7 yields \begin{align*} & |A_h(\tilde u,w_h) - f_h(w_h)| \\ & \quad = |A_h(\tilde u,w_h) - f_h(w_h) - \underbrace{\left(A(u,w_h \circ {\it{\Phi}}_h^{-1}) - f(w_h\circ {\it{\Phi}}_h^{-1})\right)}_{=0}| \\ & \quad \leq |A_h(\tilde u, w_h) - A(u, w_h \circ {\it{\Phi}}_h^{-1})| + |f_h(w_h) - f(w_h \circ {\it{\Phi}}_h^{-1})| \end{align*} which completes the proof. □ Remark 5.9 In practice, we will commit an additional variational crime due to the numerical integration of integrands as in (4.5), which are in general not polynomial. We briefly discuss how to incorporate the related additional error terms in the error analysis without going into all the technical details. The procedure is essentially the same as in Sections 25–29 in Ciarlet (1991). We consider $$A_h^q$$ and $$f_h^q$$ the bilinear and linear forms which approximate $$A_h$$ and $$f_h$$ with numerical integration. The solution obtained by quadrature is denoted as $$u_h^q \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, which satisfies $$A_h^q(u_h^q,v_h) = f_h^q(v_h)$$ for all $$v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. For the numerical integration, we consider quadrature rules that have a certain exactness degree on the (cut) elements in the reference geometries $${\it{\Omega}}^{\text{lin}}_i$$ and $${\it{\Gamma}}^{\text{lin}}$$. For the integral in (4.5), this would be a quadrature of exactness degree $$2k-2$$. The additional terms $$D{{\it{\Theta}}_h}^{-T}$$ and $$\mathrm{det}(D{{\it{\Theta}}_h})$$ stemming from the transformation are close to $${\rm id}$$ and $$1$$, respectively, cf. the estimates in Lemma 3.14. As a consequence, we have that $$A_h(u,u) \sim A_h^q(u,u)$$ on $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ and thus we get coercivity of $$A_h^q(\cdot,\cdot)$$ w.r.t. $$\|\cdot\|_h$$ on $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. The arguments in the proof of the Strang Lemma 5.8 apply with $$A_h(\cdot,\cdot)$$ replaced by $$A_h^q(\cdot,\cdot)$$. Hence, we have to bound consistency terms as in (5.17) with $$A_h(\cdot,\cdot)$$, $$f_h$$ replaced by $$A_h^q(\cdot,\cdot)$$ and $$f_h^q$$, respectively. This can be done by combining the techniques used in the proof of Lemma 5.10 below with the ones from Ciarlet (1991). 5.4 Consistency errors We derive bounds for the consistency error terms on the right-hand side in the Strang estimate (5.17). Lemma 5.10 Let $$u \in {V_{\text{reg}}}$$ be a solution of (1.1). We assume $$f \in H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ and the data extension source term $$f_h$$ in (4.4) satisfies $$\|f_h\|_{H^{1,\infty}({\it{\Omega}}_{1,h}\cup {\it{\Omega}}_{2,h})} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$. The following estimates hold for $$w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: \begin{align} |A(u,w_h\circ {\it{\Phi}}_h^{-1}) - A_h(u\circ {\it{\Phi}}_h,w_h)| & \lesssim h^k \Vert u \Vert_{H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \Vert w_h \Vert_h \label{p1} \\ \end{align} (5.19a) \begin{align} |f(w_h\circ {\it{\Phi}}_h^{-1})-f_h(w_h)| & \lesssim h^{k} \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \| w_h \Vert_h \label{p2}. \end{align} (5.19b) Proof. The proof, which is elementary, is given in the Appendix A. □ 5.5 Approximation error In this section, we derive a bound for the approximation error, i.e., the first term on the right-hand side in the Strang estimate (5.17). For this, we use the curved finite element space $$V_{h,{\it{\Psi}}}=\{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^k\,\}$$ introduced in (3.36) and the corresponding optimal interpolation operator $${\it{\Pi}}_h$$, cf. (3.37). We also use the corresponding unfitted finite element space $$V_{h,{\it{\Psi}}}^{\it{\Gamma}}:=\{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^{\it{\Gamma}}\,\}$$, cf. (4.1). Lemma 5.11 For $$u \in H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ the following holds: $$\label{approxerrro} \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert u\circ {\it{\Phi}}_h - v_h \Vert_h =\inf_{v_h \in V_{h,{\it{\Psi}}}^{\it{\Gamma}}} \Vert (u- v_h)\circ {\it{\Phi}}_h \Vert_h \lesssim h^k \|u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}.$$ (5.20) Proof. The identity in (5.20) follows from the definitions of the spaces $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, $$V_{h,{\it{\Psi}}}^{\it{\Gamma}}$$. The analysis is along the same lines as known in the literature (e.g., Hansbo & Hansbo, 2002; Reusken, 2008). Let $${\it{\Pi}}_h$$ be the interpolation operator in $$V_{h,{\it{\Psi}}}$$ and $$R_i$$ the restriction operator $$R_iv:=v_{|{\it{\Omega}}_{i}}$$. We use the bounded linear extension operators $$E_i:\, H^{k+1}({\it{\Omega}}_i) \to H^{k+1}({\it{\Omega}})$$, cf., for example, Theorem II.3.3 in Galdi (2011) and define $$u_i^e := E_i u_i, i=1,2$$. We define $$v_h := \sum_{i=1}^2 R_i {\it{\Pi}}_h u_i^e \in V_{h,{\it{\Psi}}}^{\it{\Gamma}}$$. Note that due to (3.37) we have a optimal interpolation error bounds for $$u_i^e \in H^{k+1}({\it{\Omega}})$$. We get: \begin{align*} |(u - v_h)\circ {\it{\Phi}}_h|_1^2& \lesssim \sum_{i=1}^2 \alpha_i \|\nabla (u_i - R_i {\it{\Pi}}_h u_i^e)\|_{L^2({\it{\Omega}}_{i})}^2 \\ & \lesssim \sum_{i=1}^2 \|\nabla (u_i^e - {\it{\Pi}}_h u_i^e)\|_{L^2({\it{\Omega}})}^2 \lesssim h^{2k} \sum_{i=1}^2 \| u_i^e \|_{H^{k+1}({\it{\Omega}})}^2. \end{align*} This yields the desired bound for the $$|\cdot|_1$$ part of the norm $$\|\cdot\|_h$$. As in Hansbo & Hansbo (2002), we obtain \begin{align*} \Vert [\hspace{-2pt}[{(u- v_h)\circ {\it{\Phi}}_h}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} &= \left(\bar \alpha / h \right)^\frac12 \|[\hspace{-2pt}[{(u - v_h)\circ {\it{\Phi}}_h}]\hspace{-2pt}]\|_{L^2({\it{\Gamma}}_h)} \lesssim h^{- \frac12} \sum_{i=1}^2 \| u_i^e - {\it{\Pi}}_h u_i^e \|_{L^2({\it{\Gamma}})} \\ & \lesssim \sum_{i=1}^2\Big(h^{-1} \| u_i^e - {\it{\Pi}}_h u_i^e \|_{L^2({\it{\Omega}})} +\| u_i^e - {\it{\Pi}}_h u_i^e \|_{H^1({\it{\Omega}})}\Big) \\ & \lesssim h^k \sum_{i=1}^2 \| u_i^e \|_{H^{k+1}({\it{\Omega}})} \lesssim h^k \sum_{i=1}^2 \| u_i \|_{H^{k+1}({\it{\Omega}}_i)}. \end{align*} Using very similar arguments, cf. (Hansbo & Hansbo, 2002), the same bound can be derived for the term $$\Vert \{\!\!\{{(u- v_h)\circ {\it{\Phi}}_h}\}\!\!\}\Vert_{-\frac12,h,{\it{\Gamma}}_h}$$. Thus, we get $$\|(u-v_h)\circ {\it{\Phi}}_h\|_h \lesssim h^k \| u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$, which completes the proof. □ 5.6 Discretization error bounds Based on the Strang estimate and the results derived in the subsections above, we immediately obtain the following theorem, which is the main result of this article. Theorem 5.12 Let $$u$$ be the solution of (1.1) and $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ the solution of (4.2). We assume that $$u\in H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ and the data extension $$f_h$$ satisfies the condition in Lemma 5.10. Then, the following holds: $$\label{main1} \Vert u\circ {\it{\Phi}}_h - u_h \Vert_h \lesssim h^k(\| u \|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} + \|f\|_{H^{1,\infty}({\it{\Omega}}_1\cup{\it{\Omega}}_2)}).$$ (5.21) Proof. The result directly follows from the Strang Lemma 5.8 and the bounds derived in Lemmas 5.10 and 5.11. □ 6. Discussion and outlook We presented a rigorous error analysis of a high-order unfitted FEM applied to a model interface problem. The key component in the method is a parametric mapping $${{\it{\Theta}}_h}$$, which transforms the piecewise linear planar interface approximation $${{\it{\Gamma}}^{\mathrm{lin}}}$$ to a higher-order approximation $${\it{\Gamma}}_h$$ of the exact interface $${\it{\Gamma}}$$. This mapping is based on a new local level set-based mesh transformation combined with an extension technique known from isoparametric finite elements. The corresponding isoparametric unfitted finite element space is used for discretization, and combined with a standard Nitsche technique for enforcing continuity of the solution across the interface in a weak sense. The discretization error analysis is based on a Strang Lemma. For handling the geometric error, a suitable bijective mapping $${\it{\Phi}}_h$$ on $${\it{\Omega}}$$ is constructed, which maps the numerical interface $${\it{\Gamma}}_h$$ to the exact interface $${\it{\Gamma}}$$, and is sufficiently close to the identity. The construction of this mapping is based on the same approach as used in the construction of $${{\it{\Theta}}_h}$$. The main results of the article are the optimal high-order ($$H^1$$-norm) error bounds given in Section 5.6. The derivation of an optimal order $$L^2$$-norm error bound, based on a duality argument, will be presented in a forthcoming article. For this, we need an improvement of the consistency error bound in (5.19a), which is of the order $$\mathcal{O}(h^k)$$, to a bound of order $$\mathcal{O}(h^{k+1})$$. The methodology presented in this article, especially the parametric mapping and the error analysis of the geometric errors, can also be applied in other settings with unfitted FEMs, for example, fictitious domain methods (Lehrenfeld, 2016b), unfitted FEM for Stokes interface problems (Lederer et al., 2016) and trace FEMs for surface PDEs (Grande et al., 2016). Appendix A A.1 Proof of Lemma 3.5 First, we prove the existence of a unique solution $$d_h(x)$$ of (3.12). For $$|\alpha| \leq \alpha_0 h$$, with $$\alpha_0 >0$$ fixed, we introduce the polynomial $p(\alpha):= \mathcal{E}_T \phi_h(x + \alpha G_h(x)) - \hat \phi_h (x)$ for a fixed $$x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. From (2.4) it follows that $$\label{estt} | \hat \phi_h (x) - \phi(x)| \lesssim h^2.$$ (A.1) Furthermore, with $$y:=x +\alpha G_h(x)$$, we have $$\|x-y\|_2 \lesssim h$$, and using a Taylor expansion and (2.3), we obtain (for $$\xi \in (0,1)$$) $$\label{Taylor} \begin{split} \big| (\mathcal{E}_T \phi_{h})(y)-\phi(y)\big| = & \ \Bigg| \sum_{m=0}^k \frac{1}{m!} D^m(\mathcal{E}_T \phi_{h}-\phi)(x)(y-x, \ldots,y-x) \\ & \ + \frac{1}{(k+1)!}D^{k+1}\phi(x+\xi(y-x))(y-x,\ldots,y-x)\Bigg| \\ \lesssim & \sum_{m=0}^k |\phi_h -\phi|_{m, \infty,T} \|y-x\|_2^m + \|y-x\|_2^{k+1} \lesssim h^{k+1}. \end{split}$$ (A.2) Thus, we get $$p(\alpha)= \phi(x+ \alpha G_h(x))- \phi(x) +\mathcal{O}(h^2)$$, where the constant in $$\mathcal{O}(\cdot)$$ is independent of $$x$$. Using Lemma 3.3 and (2.2), we obtain $$\label{lk} p(\alpha)= \alpha \|\nabla \phi(x)\|_2^2 + \mathcal{O}(h^2).$$ (A.3) Hence, there exists $$h_0 >0$$, such that for all $$0< h\leq h_0$$ and all $$x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, the equation $$p(\alpha)=0$$ has a unique solution $$\alpha =:d_h(x)$$ in $$[-\alpha_0 h,\alpha_0 h]$$. The result in (3.13b) follows from (A.3). The smoothness property, $$d_h \in C^\infty(T)$$, $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, follows from the fact that $$G_h$$ and $$\hat \phi_h$$ are polynomials on $$T$$, and $$\mathcal{E}_T \phi_h$$ is a global polynomial. We note that for each vertex $$x_i$$ of $$T \in {\mathcal T}$$, we have $$\mathcal{E}_T \phi_h(x_i) = \phi_h(x_i) = {\hat \phi_h}(x_i)$$ and it follows that $$d_h(x_i)=0$$ solves (3.12), and hence (3.13a) holds. We finally consider (3.13c). We differentiate the relation (3.12) on $$T$$. We skip the argument $$x$$ in the notation and use $$y_h:=x+d_h(x) G_h(x)$$: $$\label{kjj} \nabla(\mathcal{E}_T\phi_h)(y_h) + \nabla d_h \nabla (\mathcal{E}_T\phi_h)(y_h)^TG_h + d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h) = \nabla \hat \phi_h.$$ (A.4) Using a Taylor expansion as in (A.2) we get $$|\nabla(\mathcal{E}_T\phi_h)(y_h)-\nabla \phi(y_h)| \lesssim h^k$$. Using (3.13b), (3.11) and rearranging terms in (A.4), we get (for $$k \geq 2$$): $\nabla d_h \big(\|\nabla \phi\|_2^2 + \mathcal{O}(h^2)\big)= \nabla \hat \phi_h - \nabla(\mathcal{E}_T\phi_h)(y_h)-d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h).$ From the estimates derived above and the smoothness assumption on $$\phi$$, it easily follows that the second and third term on the right-hand side are uniformly bounded by a constant and by $$ch^2$$, respectively. We further have $$\nabla {\hat \phi_h} = \nabla \phi + \nabla {\hat \phi_h} - \nabla \phi$$ with $$\Vert \nabla {\hat \phi_h} - \nabla \phi \Vert_{\infty} \lesssim h$$ and $$\Vert \nabla \phi \Vert_{\infty} \lesssim 1$$. Combining these results completes the proof. A.2 Proof of Lemma 3.6 Recall that $${\it{\Psi}}^{\it{\Gamma}}(x)\,{=}\, x+ d(x) G(x)$$, $${\it{\Psi}}_h^{\it{\Gamma}}(x)\,{=}\,x+d_h(x)G_h(x)$$. We have $$\|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \,{\lesssim}\, h^2, \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d_h\|_{\infty,T} \,{\lesssim}\, h^2$$, cf. (3.2b) and (3.13b). Take $$x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. We start with the triangle inequality $$\label{ttt} \begin{split} \|({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}})(x)\|_2 & = \| d(x) G(x)- d_h(x)G_h(x)\|_2 \\ & \leq |d(x)-d_h(x)| \|G\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + |d_h(x)| \, \|G_h(x)-G(x)\|_2. \end{split}$$ (A.5) The second term on the right-hand side is uniformly bounded by $$h^{k+2}$$ due to the result in Lemma 3.3. We have by construction $$\label{phiphih} \phi(x+ d(x) G(x))= \hat\phi_h(x)=(\mathcal{E}_T\phi_{h}) (x+ d_h(x) G_h(x)), x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (A.6) Using this and (2.2) we get \begin{align*} |d(x)-d_h(x)| & \sim \big| \phi(x+ d(x) G(x)) -\phi(x+ d_h(x) G(x))\big| \\ & =\big| (\mathcal{E}_T \phi_{h})(x+ d_h(x) G_h(x))-\phi(x+ d_h(x) G(x))\big| \\ & \leq \big| (\mathcal{E}_T \phi_{h}) (x+ d_h(x) G_h(x))-\phi(x+ d_h(x) G_h(x))\big| \\ & \quad +\big|\phi(x+ d_h(x) G_h(x))- \phi(x+ d_h(x) G(x))\big|. \end{align*} Using the estimate in (A.2), it follows that the first term on the right-hand side is uniformly bounded by $$h^{k+1}$$. The second term on the right-hand side is uniformly bounded by $\big|\phi(x+ d_h(x) G_h(x))- \phi(x+ d_h(x) G(x))\big| \lesssim |d_h(x)| \, \|G_h(x)- G(x)\|_2 \lesssim h^{k+2}.$ Collecting these results we obtain $$\label{NN} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d-d_h\|_{\infty,T} \lesssim h^{k+1},$$ (A.7) which yields the desired bound for the first term on the right-hand side in (A.5). Hence, the bound for the first term in (3.16) is proven. It remains to bound the derivatives. We take the derivative in the equation (A.6). To simpifly the notation, we drop the argument $$x$$ and set $$y:=x+d(x)G(x)$$, $$y_h:=x+d_h(x)G_h(x)$$. This yields $$\label{kj} \begin{split} & \nabla \phi(y) + \nabla d \nabla \phi(y)^T \nabla \phi + d DG^T \nabla \phi(y) \\ &\quad = \nabla(\mathcal{E}_T\phi_h)(y_h) + \nabla d_h \nabla (\mathcal{E}_T\phi_h)(y_h)^T G_h + d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h). \end{split}$$ (A.8) Using a Taylor expansion as in (A.2) we get $$\label{Tay} \|\nabla(\mathcal{E}_T\phi_h)(y_h)-\nabla \phi(y_h)\|_2 \lesssim h^k.$$ (A.9) Furthermore, $$\|\nabla \phi(y)- \nabla \phi(x)\|_2 \lesssim h^2$$, $$\|\nabla \phi(y_h)- \nabla \phi(x)\|_2 \lesssim h^2$$ and $$\|G_h(x)-\nabla \phi(x)\|_2 \lesssim h$$. Using these results and (3.11) and rearranging terms in (A.8), we get \begin{align*} & \big(\nabla d - \nabla d_h\big)\big(\|\nabla \phi\|_2^2 + \mathcal{O}(h)\big) \\ &\quad = \big(\nabla (\mathcal{E}_T\phi_h)(y_h)- \nabla \phi(y)\big) +\big(d_h DG_h^T \nabla(\mathcal{E}_T\phi_h)(y_h) - d DG^T \nabla \phi(y) \big). \end{align*} Using (3.11) and (A.7), we get $$|y-y_h| \lesssim h^{k+1}$$. Using this and (A.9), we obtain for the first term on the right-hand side the uniform bound $$c h^k$$. For the second term, we obtain the same uniform bound using (A.9), (A.7), (3.11) and $$\max_{T \in {\mathcal T}}\|d_h\|_{\infty,T} \lesssim h^2$$, $$\|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^2$$. This yields $$\label{re1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|\nabla d - \nabla d_h\|_{\infty,T} \lesssim h^k.$$ (A.10) Using this, (A.7) and the results in Lemmas 3.3 and 3.5, we obtain the estimates \begin{align*} & \Vert D ({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}) \Vert_{\infty,T} = \Vert D (d G - d_h G_h) \Vert_{\infty,T} \\ &\quad \lesssim \Vert \nabla (d-d_h) \Vert_{\infty, T} \Vert G \Vert_{\infty,T} + \Vert \nabla d_h \Vert_{\infty, T} \Vert G - G_h \Vert_{\infty, T} \\ & \qquad + \|d-d_h\|_{\infty, T} \Vert D G \Vert_{\infty,T} + \|d_h\|_{\infty, T} \Vert D(G-G_h) \Vert_{\infty, T} \lesssim h^k, \end{align*} which are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. A.3 Proof of Lemma 3.10 Recall that $$\mathcal{E}^{F\rightarrow T} w := \mathcal{E}^{\hat F\rightarrow \hat T} (w \circ \hat {\it{\Phi}}_F^{-1}) \circ {\it{\Phi}}_T^{-1}$$, $$\hat{w} = w \circ \hat {\it{\Phi}}_F^{-1}$$. We transform from $$T$$,$$F$$ to the corresponding reference element and face, and use $$\Vert D^n \mathcal{E}^{F \rightarrow T} w \Vert_{\infty, T} \lesssim h^{-n} \Vert D^n \mathcal{E}^{\hat F \rightarrow \hat T} \hat w \Vert_{\infty, \hat{T}}, \quad \Vert D^r \hat{w} \Vert_{\infty, \hat{F}} \lesssim h^r \Vert D^r w \Vert_{\infty, F}.$$ From this it follows that it suffices to prove $$\label{transfeq} \Vert D^n \mathcal{E}^{\hat F\rightarrow \hat T} \hat{w}_T \Vert_{\infty,\hat{T}} \lesssim \sum_{r=n}^{k+1} \Vert D^r \hat{w} \Vert_{\infty, \hat{F}}, \quad \forall \hat w \in C_0^{k+1}(\hat F), n=0,\ldots,k+1.$$ (A.11) Recall that $$\label{defE} \mathcal{E}^{\hat{F}\rightarrow \hat{T}} \hat w :=\omega^{k+1} A_k^\ast(\hat w)\circ Z+ \sum_{l=2}^k \omega^l A_l(\hat{w}) \circ Z,$$ (A.12) with $$A_l^\ast := {\rm id} - {\it{\Lambda}}_l, \quad A_l := {\it{\Lambda}}_l - {\it{\Lambda}}_{l-1} = - A_{l}^\ast + A_{l-1}^\ast$$, $${\it{\Lambda}}_l: C(\hat{F}) \rightarrow \mathcal{P}^l(\hat{F})$$ the interpolation operator, and $$\omega$$, $$Z$$ as in (3.18). The proof uses techniques also used in Bernardi (1989). We apply a standard Bramble–Hilbert argument, which yields $$\label{alstar} \Vert D^r (A_l^\ast \hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}}, \quad 0\leq r \leq l+1 \leq k+1,$$ (A.13) and thus, by a triangle inequality, $$\label{lstar} \Vert D^r (A_l \hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}} + \Vert D^{l} \hat{w} \Vert_{\infty,\hat{F}}, \quad 0\leq r \leq l+1 \leq k+1, l \geq 2.$$ (A.14) For $$Z$$, we have $$\Vert D^m Z \Vert_{\infty,\hat{T}} \lesssim \omega^{-m}, m \geq 1$$, cf. (Bernardi, 1989, Lemma 6.1). With a higher-order multivariate chain rule, cf. (Ciarlet & Raviart, 1972), we further have \begin{align} \Vert D^m A_l^\ast(\hat{w}) \circ Z \Vert_{\infty,\hat{T}} & \lesssim \sum_{r=1}^m \underbrace{\Vert D^r A_l^\ast(\hat{w}) \Vert_{\infty,\hat{F}}}_{\lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty, \hat{F}}} \sum_{\alpha \in \mathcal{J}(r,m)} \underbrace{\prod_{q=1}^m \Vert D^q Z \Vert^{\alpha_q}}_{\omega^{-m}} \\ \end{align} (A.15) \begin{align} & \lesssim \Vert D^{l+1}\hat w \Vert_{\infty,\hat{F}} \, \omega^{-m}, \quad 1 \leq m \leq l+1\leq k+1, \label{kll} \\ \text{where} \quad \mathcal{J}(r,m) & := \left\{\alpha = (\alpha_1,..,\alpha_r) \mid \sum_{q=1}^m \alpha_q = r, \sum_{q=1}^m q \alpha_q = m\right\}\!, \nonumber \end{align} (A.16) and for $$m=0$$ $$\Vert A_l^\ast(\hat{w}) \circ Z \Vert_{\infty,\hat{T}} = \Vert A_l^\ast(\hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}}.$$ Similarly, using $$D^mA_l(\hat w)=0$$ for $$m >l$$, $\Vert D^m A_l (\hat{w}) \circ Z \Vert_{\infty,\hat{T}} \lesssim \left(\Vert D^{l+1} \hat{w} \Vert_{\infty, \hat{F}} + \Vert D^{l} \hat{w} \Vert_{\infty, \hat{F}} \right) \omega^{-m}, 0 \leq m, l\leq k+1, l \geq 2.$ Using a Bramble–Hilbert argument, Leibniz formula and the bound derived in (A.16) we get, for $$n \leq k+1$$, \begin{align*} \|D^n(\omega^{k+1} A_k^\ast(\hat w)\circ Z)\|_{\infty,\hat F} & \lesssim \sum_{r=0}^{n} \omega^{k+1-r} \underbrace{|D \omega|^r}_{\lesssim 1} \Vert D^{n-r} (A_k^\ast(\hat w \circ Z)) \Vert_{\infty,\hat T} \\ & \lesssim \sum_{r=0}^{k+1} \omega^{k+1-r} \|D^{k+1}\hat w\|_{\infty,\hat F}\,\omega^{-(n-r)} \lesssim \|D^{k+1}\hat w\|_{\infty,\hat F}, \end{align*} which yields the desired bound for the first term on the right-hand side in (A.12). We now consider one term $$\omega^l A_l(\hat w)\circ Z$$, $$2 \leq l \leq k$$, from the second term on the right-hand side in (A.12). This is a polynomial of degree $$l$$, hence $$D^n(\omega^l A_l(\hat w)\circ Z) =0$$ for $$n >l$$. We have to consider only $$n \leq l \leq k$$. With similar arguments as above we get: \begin{align} \Vert D^n (\omega^l A_l(\hat w)\circ Z) \Vert_{\infty,\hat T} & \lesssim \sum_{r=0}^n \omega^{l-r} |D \omega|^r \Vert D^{n-r} (A_l(\hat w \circ Z)) \Vert_{\infty,\hat T} \\ \end{align} (A.17) \begin{align} &\lesssim \sum_{r=0}^{k+1} \underbrace{\omega^{l-r} \omega^{-(n-r)}}_{\lesssim 1 (l \geq n)} (\Vert D^l \hat w \Vert_{\infty,\hat F} + \Vert D^{l+1} \hat w \Vert_{\infty,\hat F}). \end{align} (A.18) Summing over $$l=2, \ldots, k$$ completes the proof. A.4 Proof of Lemma 5.10 We first consider the terms $$a_h(\cdot,\cdot)$$, $$a(\cdot,\cdot)$$ in $$A_h(\cdot,\cdot)$$ and $$A(\cdot,\cdot)$$, respectively, split the bilinear forms into its contributions from the two subdomains, cf. (4.5), for instance $$a_h(\cdot,\cdot) = a_h^1(\cdot,\cdot) + a_h^2(\cdot,\cdot)$$, and consider one part $$a_h^i(\cdot,\cdot)$$, $$i=1,2$$. We introduce $$\tilde w_h:= w_h \circ {\it{\Phi}}_h^{-1}$$ and $$\tilde u:= u \circ {\it{\Phi}}_h$$, and compare the reference formulation without geometrical errors \begin{align*} a^i(u, \tilde w_h) &= \alpha_i \int_{{\it{\Omega}}_i} \nabla u \cdot \nabla \tilde w_h \, {\rm d}x = \alpha_i \int_{{\it{\Omega}}_{i,h}} \det(D {\it{\Phi}}_h) D {\it{\Phi}}_h^{-T} \nabla \tilde u \cdot D {\it{\Phi}}_h^{-T} \nabla w_h \, {\rm d}y \end{align*} with the discrete variant \begin{align*} a_h^i(\tilde u,w_h) & = \alpha_i \int_{{\it{\Omega}}_{i,h}} \nabla \tilde u \cdot \nabla w_h \, {\rm d}y. \end{align*} From this we get $|a^i(u, \tilde w_h) - a_h^i(\tilde u,w_h)| = \alpha_i \left| \int_{{\it{\Omega}}_{i,h}} \nabla \tilde u^T \cdot C \nabla w_h \, {\rm d}y\right| \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\tilde u\|_{H^1({\it{\Omega}}_{i,h})}\|w_h\|_h,$ with $$C:= \det(A) (A^TA)^{-1}-I$$, $$A:=D{\it{\Phi}}_h$$. From Lemma 5.5, we have $$\|A-I\|_{\infty,{\it{\Omega}}} \,{\lesssim}\, h^k \text{and} \|A^{-1}-I\|_{\infty,{\it{\Omega}}} \,{\lesssim}\, h^k$$, which implies $$\Vert \mathrm{det}(A) - 1 \Vert_{\infty,{\it{\Omega}}} \lesssim h^k$$. We thus obtain the bound $$\Vert C \Vert_{\infty,{\it{\Omega}}} \lesssim \Vert \mathrm{det}(A) - 1 \Vert_{\infty,{\it{\Omega}}} + \Vert A - I \Vert_{\infty,{\it{\Omega}}} + \Vert A^{-1} - I \Vert_{\infty,{\it{\Omega}}} \lesssim h^k.$$ Combining these estimates results in $$\label{est1} |a^i(u, \tilde w_h) - a_h^i(\tilde u,w_h)| \lesssim h^k \Vert \tilde u \Vert_{H^1({\it{\Omega}}_{i,h})} \Vert w_h \Vert_h.$$ (A.19) With similar arguments we obtain the estimate $$\Vert \tilde u \Vert_{H^{1}({\it{\Omega}}_{i,h})} \sim \Vert u \Vert_{H^{1}({\it{\Omega}}_i)}.$$ We now consider the terms $$N_h^c(\tilde u, w_h)$$ and $$N^c(\tilde u, w_h)$$. We use a relation between $$n_{{\it{\Gamma}}_h}(x)=:n_{{\it{\Gamma}}_h}$$ and $$n_{\it{\Gamma}}({\it{\Phi}}_h(x))$$, $$x \in {\it{\Gamma}}_h \cap {{\it{\Theta}}_h}(T)={{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}_T})$$. Let $$t_1, \ldots, t_{d-1}$$ be an orthonormal basis of $$n_{{\it{\Gamma}}_h}^\perp$$. The tangent space to $${\it{\Gamma}}= {\it{\Phi}}_h({\it{\Gamma}}_h)$$ at $${\it{\Phi}}_h(x)$$ is given by $${\rm span}\{\, D{\it{\Phi}}_h(x)t_j | 1 \leq j \leq d-1\,\}=n_{\it{\Gamma}}({\it{\Phi}}_h(x))^\perp$$. Hence, $$n_{\it{\Gamma}}({\it{\Phi}}_h(x)) \in {\rm span}\{\, D{\it{\Phi}}_h(x)^{-T}n_{{\it{\Gamma}}_h}\,\}$$ holds. This yields $$\label{traans} n_{\it{\Gamma}} \circ {\it{\Phi}}_h = \frac{D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h}}{\Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2}.$$ (A.20) In the transformation from $${\it{\Gamma}}_h$$ to $${\it{\Gamma}}$$, we get the factor $$\det(D{\it{\Phi}}_h) \Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2$$ from the change in the measure, cf. (3.40). The term $$\Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2$$ is the same as the denominator in the change of normal directions (A.20). Therefore, these terms cancel and simplify the formulas below. With the same matrix $$C$$ as used above, we get \begin{align} & \big| N^c(u, \tilde w_h) - N_h^c(\tilde u, w_h) \big| \nonumber \\ &\quad = \left|\int_{{\it{\Gamma}}_h} \det (D {\it{\Phi}}_h) \, D{\it{\Phi}}_h^{-T} \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} \cdot D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \ [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s - \int_{{\it{\Gamma}}_h} \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} \cdot n_{{\it{\Gamma}}_h} \ [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s\right| \nonumber \\ &\quad = \left| \int_{{\it{\Gamma}}_h} n_{{\it{\Gamma}}_h}^T \cdot C \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s\right| \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\}\|_{-\frac12,h,{\it{\Gamma}}_h} \|[\hspace{-2pt}[{w_h}]\hspace{-2pt}]\|_{\frac12,h,{\it{\Gamma}}_h} \nonumber \\ &\quad \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\{\!\!\{{-\alpha \nabla u}\}\!\!\}\|_{-\frac12,h,{\it{\Gamma}}} \|w_h\|_h \lesssim h^{k+\frac12} \|u\|_{H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}\|w_h\|_h. \label{pl} \end{align} (A.21) Finally, we use $$[\hspace{-2pt}[{\tilde u}]\hspace{-2pt}] = 0$$ at $${\it{\Gamma}}_h$$ to note that $$N_h^c(w_h, \tilde{u})=N_h^s(\tilde{u},w_h)=N^c(\tilde w_h, u)=N^s(u, \tilde w_h)=0$$. Combining this with the bounds in (A.19), (A.21) and the definition of $$A_h(\cdot,\cdot)$$ and $$A(\cdot,\cdot)$$, we obtain the bound in (5.19a). For the other consistency term in the Strang estimate we obtain: \begin{align*} |f(\tilde w_h)-f_h(w_h)| & \leq \sum_{i=1,2} \left| \int_{{\it{\Omega}}_{i}} f_i (w_h\circ {\it{\Phi}}_h^{-1}) \, {\rm d}x - \int_{{\it{\Omega}}_{i,h}} f_{i,h} w_h\, {\rm d}x \right|\!, \end{align*} and for $$i=1,2$$ we get, using $$f_{i,h}=f_i$$ on $${\it{\Omega}}_i$$, \begin{align*} & \left| \int_{{\it{\Omega}}_{i}} f_i (w_h\circ {\it{\Phi}}_h^{-1}) \, {\rm d}x - \int_{{\it{\Omega}}_{i,h}} f_{i,h} w_h\, {\rm d}x \right| \\ &\quad \leq \left|\int_{{\it{\Omega}}_{i,h}} (\det(D{\it{\Phi}}_h) (f_{i,h} \circ {\it{\Phi}}_h) - f_{i,h}) w_h \, {\rm d}x \right|\\ &\quad \lesssim \|f_{i,h} \circ {\it{\Phi}}_h - f_{i,h} \|_{\infty,{\it{\Omega}}_{i,h}} \|w_h\|_h + {\Vert} \det(D{\it{\Phi}}_h) -1 \Vert_{\infty,{\it{\Omega}}_{i,h}} \|f_i\|_{\infty,{\it{\Omega}}_i} \|w_h\|_h. \end{align*} Finally note that $${\Vert}\det(D{\it{\Phi}}_h) -1 \Vert_{\infty,{\it{\Omega}}} \lesssim h^k$$ and $$\|f_h\|_{H^{1,\infty}({\it{\Omega}}_{1,h}\cup {\it{\Omega}}_{2,h})} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$, and thus we get: $\|f_{i,h} \circ {\it{\Phi}}_h - f_{i,h}\|_{\infty,{\it{\Omega}}_{i,h}} \leq \|f_{i,h}\|_{H^{1,\infty}({\it{\Omega}}_{i,h})} \|{\it{\Phi}}_h - {\rm id}\|_{\infty, {\it{\Omega}}} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1\cup{\it{\Omega}}_2)} h^{k+1}$ which completes the proof of (5.19b). 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( 2010 ) An unfitted hp-interface penalty finite element method for elliptic interface problems . Preprint arXiv:1007.2893 . © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

# Analysis of a high-order unfitted finite element method for elliptic interface problems

, Volume Advance Article (3) – Aug 18, 2017
37 pages

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Oxford University Press
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0272-4979
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1464-3642
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10.1093/imanum/drx041
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### Abstract

Abstract In the context of unfitted finite element discretizations, the realization of high-order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method that achieves a high-order approximation of the geometry for domains that are implicitly described by smooth-level set functions. The method is based on a parametric mapping, which transforms a piecewise planar interface reconstruction to a high-order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping, are easy to implement. In this article, we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high-order discretization. The theoretical results are confirmed in numerical experiments. 1. Introduction 1.1 Motivation In recent years, there has been a strong increase in the research on development and analysis of unfitted finite element methods (FEMS). In unfitted FEMs, a geometry description, which is separated from the computational mesh, is used to provide a more flexible handling of the geometry compared with traditional conforming mesh descriptions. Often a level set function is used to describe geometries implicitly. Recently, significant progress has been made in the construction, analysis and application of methods of this kind (see, for instance Hansbo & Hansbo, 2002; Groß & Reusken, 2007; Olshanskii et al., 2009; Fries & Belytschko, 2010; Burman & Hansbo, 2012; Burman et al., 2014). Despite these achievements, the development and rigorous error analysis of high-order accurate unfitted FEMs is still challenging. This is mainly due to the fact that efficient, highly accurate numerical integration on domains that are implicitly described is not straight forward. In the recent article by Lehrenfeld (2016a), a new approach based on isoparametric mappings of the underlying mesh (outlined in Section 1.4 below) has been introduced, which allows for an efficient, robust and highly accurate numerical integration on domains that are described implicitly by a level set function. The main contribution of this article is an error analysis of this method applied to a model interface problem. 1.2 Literature We review the state of the art with respect to higher-order geometry approximations in the literature to put the approach proposed in Lehrenfeld (2016a) and analysed in this article into its context. For discretizations based on piecewise linear (unfitted) finite elements, a numerical integration approach which is second order accurate suffices to preserve the overall order of accuracy. This is why an approximation with piecewise planar geometries is an established approach. Piecewise planar approximations allow for a tesselation into simpler geometries, e.g., simplices, on which standard quadrature rules are applicable. Hence, a robust realization is fairly simple. It can be applied to quadrilateral and hexahedral meshes, as in the marching cube algorithm (Lorensen & Cline, 1987) as well as to simplex meshes, cf. (among others) Lehrenfeld (2015b, Chapter 4), Mayer et al. (2009), Nærland (2014, Chapter 5) for triangles and tetrahedra and Lehrenfeld (2015a), Lehrenfeld (2015b, Chapter 4) for 4-prisms and pentatopes (4-simplices). Such strategies are used in many simulation codes, e.g. (Engwer & Heimann, 2012; Burman et al., 2014; Renard & Pommier, 2014; Groß et al., 2002; Carraro & Wetterauer, 2016) and are often combined with a geometrical refinement in the quadrature. Especially on octree-based meshes, this can be done very efficiently (Chernyshenko & Olshanskii, 2015). However, by construction, the tesselation approach is only second-order accurate, and hence limits the overall accuracy when combined with higher-order finite elements. Many unfitted discretizations based on piecewise linear finite elements have a natural extension to higher-order finite element spaces, see, for instance (Parvizian et al., 2007; Bastian & Engwer, 2009; Massjung, 2012; Johansson & Larson, 2013). It requires, however, additional new techniques to obtain also higher-order accuracy when errors due to the geometry approximation and quadrature are taken into account. We review such techniques, starting with approaches from the engineering literature. One such technique is based on moment-fitting approaches, resulting in special quadrature rules (Müller et al., 2013; Sudhakar & Wall, 2013). This approach provides (arbitrary) high-order accurate integration on implicit domains. However, the construction of these quadrature rules is fairly involved and the positiveness of quadrature weights cannot be guaranteed, which can lead to stability problems. Furthermore, as far as we know, there is no rigorous complexity and error analysis. For special cases also other techniques have been proposed in the literature to obtain high-order accurate approximations of integrals on unfitted domains. In Saye (2015), a new algorithm is presented, which can achieve arbitrary high-order accuracy and guarantee positivity of integration weights. However, it is applicable only on tensor-product elements. The approach is based on the idea of locally interpreting the interface as a graph over a hyperplane. In the community of the extended FEM (XFEM) approaches, applying a parametric mapping of the sub-triangulation are often used (e.g., Cheng & Fries, 2010; Dréau et al., 2010; Fries & Omerovi, 2015). The realization of such strategies is technically involved, especially in three (or higher) dimensions. Moreover, ensuring robustness (theoretically and practically) of these approaches is difficult. Concerning the finite cell method (Parvizian et al., 2007), which is also a (high-order) unfitted FEM, we refer to Abedian et al. (2013) for a comparison of different approaches for the numerical integration on unfitted geometries. We note that the aforementioned articles on numerical integration in higher-order unfitted FEMs, which are all from the engineering literature, only present new approaches for the numerical integration and carry out numerical convergence studies. Rigorous error analyses of these methods are not known. We now discuss techniques and corresponding rigorous error analyses that have been considered in the mathematical literature. In the recent articles (Burman et al., 2015; Boiveau et al., 2016), the piecewise planar approximation discussed before has been used and combined with a correction in the imposition of boundary values (based on Taylor expansions), to allow for higher-order accuracy in fictitious domain methods applied to the Dirichlet problems. Optimal order a priori error bounds are derived. For a higher-order unfitted discretization of partial differential equations (PDEs) on surfaces, a parametric mapping of a piecewise planar interface approximation has been developed and analysed in Grande & Reusken (2016). These are the only articles, which we know of, in which geometry errors in higher-order unfitted FEMs are included in the error analysis. We also mention relevant numerical analysis articles using low-order unfitted or higher-order geometrically conforming (‘fitted’) methods. For unfitted piecewise linear finite element discretizations of PDEs on surfaces, errors caused by geometry approximation are analysed in Deckelnick et al. (2014); Burman et al. (2016). Error analyses for fitted isoparametric finite elements for the accurate approximation of curved boundaries have been developed in the classical contributions Lenoir (1986); Bernardi (1989); Ciarlet (1991). In the context of interface problems, surface-bulk coupling isoparametric-fitted techniques are analysed in Elliott & Ranner (2013). Recently, these isoparametric fitted methods have also been used to adjust a simple background mesh to curved implicitly defined geometries in a higher-order accurate fashion (Demlow, 2009; Basting & Weismann, 2013; Gawlik & Lew, 2014; Omerović & Fries, 2017). The approach analysed in this article is similar to the above-mentioned approaches Lenoir (1986); Cheng & Fries (2010); Dréau et al. (2010); Basting & Weismann (2013); Gawlik & Lew (2014); Grande & Reusken (2016); Omerović & Fries (2017) in that it is also based on a piecewise planar geometry approximation, which is significantly improved using a parametric mapping. The important difference compared with Lenoir (1986); Basting & Weismann (2013); Gawlik & Lew (2014); Omerović & Fries (2017) is that we consider an unfitted discretization and compared with Cheng & Fries (2010); Dréau et al. (2010); Grande & Reusken (2016) is that we consider a parametric mapping of the underlying mesh rather than the sub-triangulation or only the interface. The discretization on the higher-order geometry approximation that we obtain from a parametric mapping is based on a variant of Nitsche’s method (Nitsche, 1971) for the imposition of interface conditions in nonconforming unfitted finite element spaces (Hansbo & Hansbo, 2002). To our knowledge, there is no literature in which geometrical errors for isoparametric or higher-order unfitted Nitsche-type discretizations have been considered. 1.3 The problem setting On a bounded connected polygonal domain $${\it{\Omega}} \subset \Bbb{R}^d$$, $$d=2,3$$, we consider the model interface problem \begin{align} - \mathrm{div} (\alpha_i \nabla {u}) &= \, f_i \quad \text{in} \,\, {\it{\Omega}}_i, \, i=1,2, \label{eq:ellmodel1} \\ \end{align} (1.1a) \begin{align} [\hspace{-2pt}[{{\alpha} \nabla {u}}]\hspace{-2pt}]_{{\it{\Gamma}}} \cdot n_{\it{\Gamma}} &= \, 0, \quad [\hspace{-2pt}[{{u}}]\hspace{-2pt}]_{{\it{\Gamma}}} = 0 \quad \text{on} \,\, {\it{\Gamma}}, \label{eq:ellmodel2} \\ \end{align} (1.1b) \begin{align} u &= 0 \quad \text{on} \,\, \partial \,\, {\it{\Omega}}. \end{align} (1.1c) Here, $${\it{\Omega}}_1 \cup {\it{\Omega}}_2= {\it{\Omega}}$$ is a nonoverlapping partitioning of the domain, $${\it{\Gamma}} = \bar {\it{\Omega}}_1 \cap \bar {\it{\Omega}}_2$$ is the interface and $$[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]_{{\it{\Gamma}}}$$ denotes the usual jump operator across $${\it{\Gamma}}$$. $$f_i$$, $$i=1,2$$ are domainwise described sources. In the remainder, we will also use the source term $$f$$ on $${\it{\Omega}}$$, which we define as $$f|_{{\it{\Omega}}_i} = f_i$$, $$i=1,2$$. The diffusion coefficient $$\alpha$$ is assumed to be piecewise constant, i.e., it has a constant value $$\alpha_i$$ on each subdomain $${\it{\Omega}}_i$$. We assume simplicial triangulations of $${\it{\Omega}}$$ which are not fitted to $${\it{\Gamma}}$$. Furthermore, the interface is characterized as the zero level of a given level set function $$\phi$$. 1.4 Basic idea of the isoparametric unfitted discretization The new idea of the method introduced in Lehrenfeld (2016a) is to construct a parametric mapping of the underlying triangulation based on a higher-order (i.e., degree at least 2) finite element approximation $$\phi_h$$ of the level set function $$\phi$$, which characterizes the interface. This mapping, which is easy to construct, uses information extracted from the level set function to map a piecewise planar interface approximation of the interface to a higher-order approximation, cf. the sketch in Fig. 1. Fig. 1. View largeDownload slide Basic idea of the method in Lehrenfeld (2016a): The geometry description with the level set function approximation $$\phi_h$$ is highly accurate but implicit (left). The zero level $${\it{\Gamma}}^{\text{lin}}$$ of the piecewise linear interpolation $$\hat{\phi}_h$$ has an explicit representation, but is only second order accurate (center). $${\it{\Gamma}}^{\text{lin}}$$ is mapped approximately to the implicit interface $$\{\phi_h=0\}$$ using a mesh transformation $${\it{\Theta}}_h$$, resulting in a highly accurate and explicit representation $${\it{\Gamma}}_h = {\it{\Gamma}}^{\text{lin}} \circ {\it{\Theta}}_h^{-1}$$ (right). Fig. 1. View largeDownload slide Basic idea of the method in Lehrenfeld (2016a): The geometry description with the level set function approximation $$\phi_h$$ is highly accurate but implicit (left). The zero level $${\it{\Gamma}}^{\text{lin}}$$ of the piecewise linear interpolation $$\hat{\phi}_h$$ has an explicit representation, but is only second order accurate (center). $${\it{\Gamma}}^{\text{lin}}$$ is mapped approximately to the implicit interface $$\{\phi_h=0\}$$ using a mesh transformation $${\it{\Theta}}_h$$, resulting in a highly accurate and explicit representation $${\it{\Gamma}}_h = {\it{\Gamma}}^{\text{lin}} \circ {\it{\Theta}}_h^{-1}$$ (right). A higher-order isoparametric unfitted FEM is then obtained by first formulating a higher-order discretization with respect to the low-order geometry approximation, which serves as a reference configuration. The parametric mapping is then applied to improve the accuracy of the geometry approximation. The application of the transformation also induces a change of the finite element space, which renders the space an isoparametric finite element space. This approach is geometry based and can be applied to unfitted interface or boundary value problems as well as to PDEs on surfaces. All volume and interface integrals that occur in the implementation of the method can be formulated in terms of integrals on the reference configuration, i.e., on simplices cut by the piecewise planar approximation of the interface. Hence, quadrature is straightforward. For the definition of the discretization of the model problem (1.1) on the reference geometry, we take an unfitted higher-order ‘cut’ finite element space, where the cut occurs at the piecewise planar zero level of the piecewise linear finite element approximation of $$\phi_h$$. Applying the mesh transformation to this space induces an isoparametric unfitted finite element space for the discretization of (1.1). In the same way, as in the seminal article (Hansbo & Hansbo, 2002), the continuity of the discrete solution across the (numerical) interface is enforced in a weak sense using Nitsche’s method. The resulting discrete problem has a unique solution $$u_h$$ in the isoparametric unfitted finite element space ($$h$$ is related in the usual way to the size of the simplices in the triangulation). 1.5 Content and structure of the article In this article, we present a rigorous error analysis of the method introduced in Lehrenfeld (2016a), applied to the interface problem (1.1). The main result of the article is the discretization error bound in Theorem 5.12. As a corollary of that result, we obtain for isoparametric unfitted finite elements of degree $$k \geq 1$$ an error bound of the form $h^{-\frac12} \Vert [\hspace{-2pt}[{\tilde{u}_h}]\hspace{-2pt}]_{\it{\Gamma}} \Vert_{L^2({\it{\Gamma}})} + \|u- \tilde{u}_h \|_{H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \leq c h^k\big(\|u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} + \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \big),$ with a constant $$c>0$$ that is independent of $$h$$ and the interface position within the computational mesh. $$\tilde{u}_h := u_h \circ {\it{\Phi}}_h^{-1}$$ is the mapped discrete solution, where $${\it{\Phi}}_h$$ is a smooth transformation, which maps from the approximated to the exact domains $${\it{\Omega}}_i, i=1,2$$ and is close to the identity (precise definition given in Section 5.2). Here, $$\|\cdot\|_{H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}=\|\cdot\|_{H^{1,2}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$ and $$\|\cdot\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$ are broken Sobolev norms with $$\|\cdot\|_{W({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}^2 = \sum_{i=1,2} \|\cdot\|_{W({\it{\Omega}}_i)}^2$$, $$W \in \{H^1, H^{k+1}, H^{1,\infty} \}$$. As far as we know, this is the first rigorous higher-order error bound for an unfitted FEM applied to a problem with an implicitly given interface. The article is organized as follows. In Section 2, we introduce notation and assumptions, and define a finite element projection operator. The parametric mapping $${\it{\Theta}}_h$$ is introduced in Section 3. In Section 4, the isoparametric unfitted FEM is presented and numerical examples are given. The main contribution of this article is the error analysis of the method given in Section 5. 2. Preliminaries 2.1 Notation and assumptions We introduce notation and assumptions. The simplicial triangulation of $${\it{\Omega}}$$ is denoted by $${\mathcal T}$$ and the standard finite element space of continuous piecewise polynomials up to degree $$k$$ by $$V_h^k$$. The nodal interpolation operator in $$V_h^k$$ is denoted by $$I_k$$. For the ease of presentation, we assume quasi-uniformity of the mesh, s.t. $$h$$ denotes a characteristic mesh size with $$h\sim h_T:={\rm diam}(T)$$, $$T\in{\mathcal T}$$. We assume that the smooth interface $${\it{\Gamma}}$$ is the zero level of a smooth level set function $$\phi$$, i.e., $${\it{\Gamma}}= \{\, x \in {\it{\Omega}} | \phi(x)=0\,\}$$. This level set function is not necessarily close to a distance function, but has the usual properties of a level set function: $$\label{LSdef} \|\nabla \phi(x)\| \sim 1, \,\, \|D^2 \phi(x)\| \leq c \quad \text{for all}\,\, x \,\, \text{in a neighborhood U of {\it{\Gamma}}}.$$ (2.1) We assume that the level set function has the smoothness property $$\phi \in C^{k+2}(U)$$. The assumptions on the level set function (2.1) imply the following relation, which is fundamental in the analysis below $$\label{eq1} |\phi(x+\epsilon \nabla \phi(x)) - \phi(x+\tilde \epsilon \nabla \phi(x))| \sim |\epsilon - \tilde \epsilon| \quad x \in U,$$ (2.2) for $$|\epsilon|, |\tilde \epsilon|$$ sufficiently small. As an input for the parametric mapping, we need an approximation $$\phi_h \in V_h^k$$ of $$\phi$$, and we assume that this approximation satisfies the error estimate $$\label{err2} \max_{T\in {\mathcal T}} |\phi_h - \phi|_{m,\infty,T \cap U} \lesssim h^{k+1-m},\quad 0 \leq m \leq k+1.$$ (2.3) Here $$|\cdot|_{m,\infty,T\cap U}$$ denotes the usual seminorm on the Sobolev space $$H^{m,\infty}(T\cap U)$$. Note that (2.3) holds for the nodal interpolation $$\phi_h = I_k \phi$$ and implies the estimate $$\label{err1} \|\phi_h - \phi\|_{\infty, U} + h\|\nabla(\phi_h - \phi)\|_{\infty, U} \lesssim h^{k+1}.$$ (2.4) Here, and in the remainder, we use the notation $$\lesssim$$ (and $$\sim$$ for $$\lesssim$$ and $$\gtrsim$$), which denotes an inequality with a constant that is independent of $$h$$ and of how the interface $${\it{\Gamma}}$$ intersects the triangulation $${\mathcal T}$$. This constant may depend on $$\phi$$ and on the diffusion coefficient $$\alpha$$, cf. (1.1a). In particular, the estimates that we derive are not uniform in the jump in the diffusion coefficient $$\alpha_1/\alpha_2$$. The zero level of the finite element function $$\phi_h$$ (implicitly) characterizes the discrete interface. The piecewise linear nodal interpolation of $$\phi_h$$ is denoted by $${\hat \phi_h} = I_1 \phi_h$$. Hence, $${\hat \phi_h}(x_i)=\phi_h(x_i)$$ at all vertices $$x_i$$ in the triangulation $${\mathcal T}$$. The low-order geometry approximation of the interface, which is needed in our discretization method, is the zero level of this function, $${{\it{\Gamma}}^{\mathrm{lin}}} := \{{\hat \phi_h} = 0\}$$. The corresponding subdomains are denoted as $${{\it{\Omega}}^{\mathrm{lin}}_i} = \{{\hat \phi_h} \lessgtr 0 \}$$. All elements in the triangulation $${\mathcal T}$$ which are cut by $${{\it{\Gamma}}^{\mathrm{lin}}}$$ are collected in the set $${{\mathcal T}^{{\it{\Gamma}}}} := \{T \in {\mathcal T}, T \cap {{\it{\Gamma}}^{\mathrm{lin}}} \neq \emptyset \}$$. The corresponding domain is $${{\it{\Omega}}^{{\it{\Gamma}}}} := \{x \in T, T\in {{\mathcal T}^{{\it{\Gamma}}}}\}$$. The extended set, which includes all neighbors that share at least one vertex with elements in $${{\mathcal T}^{{\it{\Gamma}}}}$$, is $${{\mathcal T}^{{\it{\Gamma}}}_+} := \{T \in {\mathcal T}, T\cap {{\it{\Omega}}^{{\it{\Gamma}}}} \neq \emptyset \}$$ with the corresponding domain $${{\it{\Omega}}^{{\it{\Gamma}}}_+} := \{x \in T, T \in {{\mathcal T}^{{\it{\Gamma}}}}\}$$. 2.2 Projection operator onto the finite element space $$V_h^k({\it{\Omega}}^{\it{\Gamma}})^d$$ In the construction of the isoparametric mapping $${\it{\Theta}}_h$$, we need a projection step from a function which is piecewise polynomial, but discontinuous (across element interfaces) to the space of continuous finite element functions. Let $$C({{\mathcal T}^{{\it{\Gamma}}}}) = \bigoplus\limits_{T\in {{\mathcal T}^{{\it{\Gamma}}}}} C(T)$$ and $$V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}}) := V_h^k|_{{{\it{\Omega}}^{{\it{\Gamma}}}}}$$. We introduce a projection operator $$P_h^{\it{\Gamma}}: C({{\mathcal T}^{{\it{\Gamma}}}})^d \rightarrow V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$. The projection operator relies on a nodal representation of the finite element space $$V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})$$. The set of finite element nodes $$x_i$$ in $${{\mathcal T}^{{\it{\Gamma}}}}$$ is denoted by $$N({{\mathcal T}^{{\it{\Gamma}}}})$$, and $$N(T)$$ denotes the set of finite element nodes associated with $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. All elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$ that contain the same finite element node $$x_i$$ form the set denoted by $$\omega(x_i)$$: \begin{align*} \omega(x_i) & := \{T \in {{\mathcal T}^{{\it{\Gamma}}}} \,{|}\, x_i \in N(T) \}, x_i \in N({{\mathcal T}^{{\it{\Gamma}}}}). \end{align*} For each finite element node, we define the local average as $$\label{average} A_{x_i}(v):= \frac{1}{|\omega(x_i)|} \sum_{T \in \omega(x_i)} v_{|T}(x_i), x_i \in N({{\mathcal T}^{{\it{\Gamma}}}}),$$ (2.5) where $$|\cdot|$$ denotes the cardinality of the set $$\omega(x_i)$$. The projection operator $$P_h^{\it{\Gamma}}: C({{\mathcal T}^{{\it{\Gamma}}}})^d \to V_h^k({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$ is defined as \begin{equation*} P_h^{\it{\Gamma}} v:= \sum_{x_i \in N({{\mathcal T}^{{\it{\Gamma}}}})} A_{x_i}(v) \, \psi_i, \quad v \in C({{\mathcal T}^{{\it{\Gamma}}}})^d, \end{equation*} where $$\psi_i$$ is the nodal basis function corresponding to $$x_i$$. This is a simple and well-known projection operator considered also in, e.g., Oswald (1993, Eqs. (25)–(26)) and Guermond & Ern (2015). Note that $$P_h^{\it{\Gamma}} w = I_k w \,\,\forall \,\, w \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d, \quad \|P_h^{\it{\Gamma}} w\|_{\infty,{\it{\Omega}}} \lesssim \max_{T \in{{\mathcal T}^{{\it{\Gamma}}}}} \| w\|_{\infty,T} \,\, \forall \,\, w \in C({{\mathcal T}^{{\it{\Gamma}}}})^d.$$ (2.6) 3. The isoparametric mapping $$\boldsymbol{{\it{\Theta}}}_{\boldsymbol{h}}$$ In this section, we introduce the transformation $${\it{\Theta}}_h \in (V_h^k)^d$$, which is a bijection on $${\it{\Omega}}$$ and satisfies $${\it{\Theta}}_h = {\rm id}$$ on $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$. This mapping is constructed in two steps. First, a local mapping $${\it{\Theta}}_h^{\it{\Gamma}}$$, which is defined on $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, is derived, and this local mapping is then extended to the whole domain. We also need another bijective mapping on $${\it{\Omega}}$$, denoted by $${\it{\Psi}}$$, which is constructed in a similar two-step procedure. This mapping $${\it{\Psi}}$$ is needed in the error analysis and in the derivation of important properties of $${\it{\Theta}}_h$$. In the higher-order FEM, which is presented in Section 4, the mapping $${\it{\Theta}}_h$$ is a key component, and the mapping $${\it{\Psi}}$$ is not used. In this section, we introduce both $${\it{\Theta}}_h$$ and $${\it{\Psi}}$$, because the construction of both mappings has strong similarities. Since the construction is not standard and consists of a two-step procedure, we outline our approach, cf. Fig. 2: We start with the relatively simple definition of the local mapping $${\it{\Psi}}^{\it{\Gamma}}$$, which has the property $${\it{\Psi}}^{\it{\Gamma}}({{\it{\Gamma}}^{\mathrm{lin}}})={\it{\Gamma}}$$ (Section 3.1), but is only defined in $${\it{\Omega}}^{\it{\Gamma}}$$. In general, this mapping cannot (efficiently) be constructed in practice. The definition of $${\it{\Psi}}^{\it{\Gamma}}$$ is slightly modified, to allow for a computationally efficient construction, which then results in the local isoparametric mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ with the property $${{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})={\it{\Gamma}}_h \approx {\it{\Gamma}}$$ (Section 3.2). $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is only defined in $${\it{\Omega}}^{\it{\Gamma}}$$. The extension of the mappings $${\it{\Psi}}^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is based on the same general procedure, which is derived from extension techniques that are standard in the literature on isoparametric FEMs (cf. Lenoir, 1986; Bernardi, 1989). This general procedure is explained in Section 3.3. The general extension procedure is applied to the mapping $${\it{\Psi}}^{\it{\Gamma}}$$, resulting in the global bijection $${\it{\Psi}}$$. Important properties of $${\it{\Psi}}$$ are derived (Section 3.4). Finally, the general extension procedure is applied to the mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, resulting in the global bijection $${{\it{\Theta}}_h}$$. Important properties of $${{\it{\Theta}}_h}$$ are derived (Section 3.5). Fig. 2. View largeDownload slide Sketch of different mesh transformations $${\it{\Psi}}^{\it{\Gamma}}$$, $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, $${\it{\Psi}}$$, $${\it{\Theta}}_h$$ and $${\it{\Phi}}_h := {\it{\Psi}} \circ {\it{\Theta}}_h^{-1}$$. Fig. 2. View largeDownload slide Sketch of different mesh transformations $${\it{\Psi}}^{\it{\Gamma}}$$, $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, $${\it{\Psi}}$$, $${\it{\Theta}}_h$$ and $${\it{\Phi}}_h := {\it{\Psi}} \circ {\it{\Theta}}_h^{-1}$$. 3.1 Construction of $${\it{\Psi}}^{\it{\Gamma}}$$ We introduce the search direction $$G:=\nabla \phi$$ and a function $$d: {{\it{\Omega}}^{{\it{\Gamma}}}} \to {\mathbb{R}}$$ defined as follows: $$d(x)$$ is the (in absolute value) smallest number such that $$\label{cond1} \phi(x + d(x) G(x))=\hat \phi_h(x) \quad \text{for} \,\, x \,\, \in {{\it{\Omega}}^{{\it{\Gamma}}}}.$$ (3.1) (Recall that $$\hat\phi_h$$ is the piecewise linear nodal interpolation of $$\phi_h$$.) We summarize a few properties of the function $$d$$. Lemma 3.1 For $$h$$ sufficiently small, the relation (3.1) defines a unique $$d(x)$$ and $$d \in C({{\it{\Omega}}^{{\it{\Gamma}}}}) \cap H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}}) \cap C^{k+1}({{\mathcal T}^{{\it{\Gamma}}}})$$. Furthermore, there holds \begin{align} |d(x_i)| & \lesssim h^{k+1} \quad \text{for all vertices $x_i$ of} \,\, T \in {{\mathcal T}^{{\it{\Gamma}}}},\label{resd4a0}\\ \end{align} (3.2a) \begin{align} \|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} &\lesssim h^2, \label{resd4a} \\ \end{align} (3.2b) \begin{align} \|d\|_{H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})} &\lesssim h, \label{resd4b} \\ \end{align} (3.2c) \begin{align} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d\|_{H^{l,\infty}(T)} &\lesssim 1, \,\, \text{for} \,\, l \leq k+1. \label{resd4c} \end{align} (3.2d) Proof. For $$|\alpha| \leq \alpha_0 h$$, with a fixed $$\alpha_0 >0$$, we introduce for a fixed $$x \in {{\it{\Omega}}^{{\it{\Gamma}}}}$$ the continuous function $g(\alpha):=\phi(x+ \alpha G(x))- \hat \phi_h(x).$ From (2.4), one obtains $$\|\hat \phi_h - \phi\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^2$$. Using this, a triangle inequality and (2.1), we get $$\label{lpp} g(\alpha)= \alpha \|\nabla \phi(x)\|_2^2 + \mathcal{O}(h^2).$$ (3.3) Hence, there exists $$h_0 >0$$ such that for all $$0< h\leq h_0$$ and all $$x\in {{\it{\Omega}}^{{\it{\Gamma}}}}$$, the equation $$g(\alpha)=0$$ has a unique solution $$\alpha =:d(x)$$ in $$[-\alpha_0 h,\alpha_0 h]$$. The result in (3.2b) follows from (3.3). Using (2.2) and (2.4) we get, for a vertex $$x_i$$ of $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, $|d(x_i)| \sim |\phi(x_i+d(x_i)G(x_i))-\phi(x_i)|= |\hat\phi_h(x_i)-\phi(x_i)|=|\phi_h(x_i)-\phi(x_i)| \lesssim h^{k+1},$ which is the estimate in (3.2a). The continuity of $$d$$ on $${{\it{\Omega}}^{{\it{\Gamma}}}}$$ follows from the continuity of $$\phi, G$$ and $$\hat \phi$$. For (3.2c), we differentiate (3.1) and write $$y = x + d(x) G(x)$$: \begin{align*} \nabla \phi(y) + \nabla d \nabla \phi(y)^T G + d (D G^T) \nabla \phi(y) & = \nabla {\hat \phi_h}(x). \end{align*} This yields \begin{align*} \nabla d (\Vert \nabla \phi \Vert_2^2 + \mathcal{O}(h^2)) + \mathcal{O}(h^2) & = \nabla {\hat \phi_h}(x) - \nabla \phi(y) = \nabla {\hat \phi_h}(x) - \nabla \phi(x) + \mathcal{O}(h^2), \end{align*} which, in combination with $$\|\nabla {\hat \phi_h} - \nabla \phi\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h$$, implies $$\Vert \nabla d(x) \Vert \lesssim h$$ a.e. in $${{\it{\Omega}}^{{\it{\Gamma}}}}$$. For $$T\in {{\mathcal T}^{{\it{\Gamma}}}}$$, we consider the function $$F(x,y) = \phi(x+y G(x))-{\hat \phi_h}(x), (x,y)\in T \times (-\alpha_0 h,\alpha_0 h)$$. The function $$d(x)=y(x)$$ solves the implicit equation $$F(x,y(x))=0$$ on $$T$$. We recall the regularity assumption $$\phi\in C^{k+2}(U)$$. From the implicit function theorem, we know that $$y \in C^{k+1}(T)$$, because $$F \in C^{k+1}\big(T \times (- \alpha_0 h, \alpha_0 h)\big)$$. From $$y \in C^{k+1}({{\mathcal T}^{{\it{\Gamma}}}})$$ and $$y \in C({{\it{\Omega}}^{{\it{\Gamma}}}})$$, we conclude $$y \in H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})$$. Note that $$D^\alpha \hat \phi_h=0$$ for $$|\alpha| \geq 2$$ and $$\|D^\alpha \hat \phi_h\|_{\infty,T} \lesssim \| \phi \|_{H^{2,\infty}(T)}$$ for $$|\alpha| \leq 1$$. Hence, $$\label{lp} \|D^\alpha_{(x,y)}F\|_{\infty, T \times (\alpha_0h, \alpha_0 h)} \lesssim \|\phi\|_{H^{l+1,\infty}(U)}\quad \text{for} \,\, |\alpha| \leq k+1.$$ (3.4) Differentiating $$F(x,y(x))=0$$ yields $$\label{eq:impl:1} D^\alpha y(x) = -D_y F(x,y(x))^{-1} D_x^{\alpha} F(x,y(x)) = -A(x) D_x^{\alpha} F(x,y(x)), \quad |\alpha|=1,$$ (3.5) with $$A(x) = S(x)^{-1}$$, $$S(x) = D_y F(x,y(x))= \nabla \phi(x+yG(x))^T\nabla \phi(x) \in [c_0,c_1]$$ with $$c_0, c_1 > 0$$ independent of $$h,x,T$$. Differentiating $$S(x) A(x) = I$$ we get $$\label{eq:impl:2} D^\alpha A(x) = - A(x)^2 D^\alpha S(x), \quad |\alpha|=1.$$ (3.6) From the relations (3.5) and (3.6), it follows that $$|D^\alpha y(x)|$$, $$|\alpha| = l$$ can be bounded in terms of $$\|A(x)\|$$ and $$|D^\alpha_{(x,y)} F(x,y(x))|$$, $$|\alpha|\leq l$$. Combining this with (3.4) proves (3.2d). □ Given the function $$dG \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d \cap H^{1,\infty}({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$, we define $$\label{psi1} {{\it{\Psi}}^{\it{\Gamma}}}(x):= x + d(x) G(x), \quad x \in {{\it{\Omega}}^{{\it{\Gamma}}}}.$$ (3.7) Note that the function $$d$$ and mapping $${\it{\Psi}}^{\it{\Gamma}}$$ depend on $$h$$, through $$\hat \phi_h$$ in (3.1). We do not show this dependence in our notation. As a direct consequence of the estimates derived in Lemma 3.1 and the smoothness assumption $$G=\nabla \phi \in C^{k+1}(U)$$, we have the following uniform bounds on (higher) derivatives of $${\it{\Psi}}^{\it{\Gamma}}$$. Corollary 3.2 The following holds: \begin{align} \|{{\it{\Psi}}^{\it{\Gamma}}} - {\rm id} \|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} +h \|D{{\it{\Psi}}^{\it{\Gamma}}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} & \lesssim h^2, \label{fistderPsi}\\ \end{align} (3.8) \begin{align} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \Vert D^l {{\it{\Psi}}^{\it{\Gamma}}} \Vert_{\infty,T} & \lesssim 1, \quad l \leq k+1.\label{fistderPsiA} \end{align} (3.9) 3.2 Construction of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ The construction of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ consists of two steps. In the first step, we introduce a discrete analogon of $${\it{\Psi}}^{\it{\Gamma}}$$ defined in (3.7), denoted by $${\it{\Psi}}_h^{\it{\Gamma}}$$. Based on this $${\it{\Psi}}_h^{\it{\Gamma}}$$, which can be discontinuous across element interfaces, we obtain a continuous transformation $${{\it{\Theta}}_h^{\it{\Gamma}}} \in C({{\it{\Omega}}^{{\it{\Gamma}}}})^d$$ by averaging with the projection operator $$P_h^{\it{\Gamma}}$$ from Section 2.2. For the construction of $${\it{\Psi}}_h^{\it{\Gamma}}$$, we need an efficiently computable good approximation of $$G=\nabla \phi$$ on $${{\mathcal T}^{{\it{\Gamma}}}}$$. For this, we consider the following two options $$\label{eq:gh} G_h(x) = \nabla \phi_h(x) \quad \text{or} \,\, G_h(x) = (P_h^{\it{\Gamma}} \nabla \phi_h)(x), \quad x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (3.10) Lemma 3.3 Both for $$G_h = P_h^{\it{\Gamma}} \nabla \phi_h$$ and $$G_h= \nabla \phi_h$$, the estimate $$\label{eqlem2} \|G_h - G\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + h \max_{T\in {{\mathcal T}^{{\it{\Gamma}}}}} \| D (G_h - G) \|_{\infty, T} \lesssim h^{k}$$ (3.11) holds with $$G = \nabla \phi$$. Proof. For $$G_h= \nabla \phi_h$$, the result is a direct consequence of (2.4). For $$G_h = P_h^{\it{\Gamma}} \nabla \phi_h$$, we use \begin{align*} \Vert P_h^{\it{\Gamma}} \nabla \phi_h - \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \leq \Vert P_h^{\it{\Gamma}} \nabla \phi_h - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + \Vert \nabla \phi - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}}, \\ \Vert P_h^{\it{\Gamma}} \nabla \phi_h - I_k \nabla \phi \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} = \Vert P_h^{\it{\Gamma}} (\nabla \phi_h - \nabla \phi) \Vert_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim \max_{T\in{{\mathcal T}^{{\it{\Gamma}}}}} \Vert \nabla \phi_h - \nabla \phi \Vert_{\infty,T}, \end{align*} and (2.4) to estimate $$\|G_h - G\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^k$$. For the derivative, we make use of (2.3) and (2.4): \begin{align*} \Vert D (G_h-G) \Vert_{\infty,T} & \leq \Vert D(G_h - I_k G) \Vert_{\infty,T} + \Vert D(G - I_k G) \Vert_{\infty,T} \\ & \lesssim \frac{1}{h} \Vert G_h - I_k G \Vert_{\infty,T} + h^{k-1} \\ & \lesssim \frac{1}{h} \Vert G_h - G \Vert_{\infty,T} + \frac{1}{h} \Vert G - I_k G \Vert_{\infty,T} + h^{k-1} \lesssim h^{k-1}, \end{align*} with estimates that are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. □ Remark 3.4 We comment on the options in (3.10). An important property of the search direction $$G_h$$ is the proximity to a continuous vector field $$G$$, which describes the direction with respect to which the interface $${\it{\Gamma}}$$ can be interpreted as a graph on $${{\it{\Gamma}}^{\mathrm{lin}}}$$. For each $$x \in {\it{\Gamma}}$$, there exists a unique $$y \in {{\it{\Gamma}}^{\mathrm{lin}}}$$ and $$d\in {\mathbb{R}}$$, s.t. $$x = y + d\, G(x)$$. The choices in (3.10) are accurate approximations of $$G(x) = \nabla \phi$$, which (locally) have the graph property for sufficiently smooth interfaces $${\it{\Gamma}}$$. Despite the discontinuities across element interfaces, the discrete search direction $$G_h = \nabla \phi_h$$ is a reasonable choice, as the distance to the continuous search direction $$G=\nabla \phi$$ is sufficiently small, cf. Lemma 3.3. Let $$\mathcal{E}_T \phi_h$$ be the polynomial extension of $$\phi_h|_T$$. We define a function $$d_h: {{\mathcal T}^{{\it{\Gamma}}}} \to [-\delta,\delta]$$, with $$\delta > 0$$ sufficiently small, as follows: $$d_h(x)$$ is the (in absolute value) smallest number such that $$\label{eq:psihmap} \mathcal{E}_T \phi_h(x + d_h(x) G_h(x)) = \hat \phi_h, \quad \text{for} \,\, x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (3.12) Clearly, this $$d_h(x)$$ is a ‘reasonable’ approximation of the steplength $$d(x)$$ defined in (3.1). We summarize a few properties of the function $$d_h$$. Lemma 3.5 For $$h$$ sufficiently small, the relation (3.12) defines a unique $$d_h(x)$$ and $$d_h \in C^\infty({{\mathcal T}^{{\it{\Gamma}}}})$$. Furthermore: \begin{gather} d_h(x_i) =0 \quad \text{for all vertices $x_i$ of $T \in {{\mathcal T}^{{\it{\Gamma}}}}$}, \label{resd1} \\ \end{gather} (3.13a) \begin{gather} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d_h\|_{\infty,T} \lesssim h^2, \label{resd4} \\ \end{gather} (3.13b) \begin{gather} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|\nabla d_h\|_{\infty,T} \lesssim 1. \label{resd6} \end{gather} (3.13c) Proof. These results can be derived using arguments very similar to the ones used in the proof of Lemma 3.1. For completeness, such a proof is given in the Appendix A, Section A.1. □ Given the function $$d_h G_h \in C({{\mathcal T}^{{\it{\Gamma}}}})^d$$, we define $$\label{eq:psih} {\it{\Psi}}_h^{\it{\Gamma}}(x) := x + d_h(x) G_h(x) \quad \text{for} \,\, x \in T \in {{\mathcal T}^{{\it{\Gamma}}}},$$ (3.14) which approximates the function $${\it{\Psi}}^{\it{\Gamma}}$$ defined in (3.7). To remove the possible discontinuities of $${\it{\Psi}}_h^{\it{\Gamma}}$$ in $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, we apply the projection to obtain $${{\it{\Theta}}_h^{\it{\Gamma}}}:= P_h^{\it{\Gamma}} {\it{\Psi}}_h^{\it{\Gamma}} = {\rm id} + P_h^{\it{\Gamma}}(d_h G_h).$$ (3.15) Using the results in the Lemmas 3.1 and 3.5, one can derive estimates on the difference $${\it{\Psi}}^{\it{\Gamma}}- {\it{\Psi}}_h^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}- {\it{\Psi}}_h^{\it{\Gamma}}$$. Such results, which are needed in the error analysis, are presented in the next two lemmas. Lemma 3.6 The estimate $$\label{estPsi} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|{\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}\|_{\infty,T} + h \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|D ({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}) \|_{\infty,T} \lesssim h^{k+1}$$ (3.16) holds. Proof. In the Appendix A, Section A.2, we give a proof that is based on the definitions of $${\it{\Psi}}_h^{\it{\Gamma}}$$, $${\it{\Psi}}^{\it{\Gamma}}$$ and the properties of $$d_h$$ and $$d$$ in Lemmas 3.5 and 3.1. □ Next, we compare $${\it{\Psi}}^{\it{\Gamma}}$$ with the isoparametric mapping $${\it{\Theta}}_h^{\it{\Gamma}}= P_h {\it{\Psi}}_h^{\it{\Gamma}}$$. Lemma 3.7 The estimate $$\sum_{r=0}^{k+1} h^r \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \lesssim h^{k+1}$$ (3.17) holds. Proof. We have with $$P_h v = I_k v$$ for $$v \in C({{\it{\Omega}}^{{\it{\Gamma}}}})$$ \begin{align*} \|{{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} & \leq \|P_h ({\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + \| I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \\ & \lesssim \| {\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}} \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} + h^{k+1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\| d G\|_{H^{k+1,\infty}(T)}. \end{align*} With (3.2d) and the smoothness of $$G$$, we have that $$\max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \| d G\|_{H^{k+1,\infty}(T)}$$ is uniformly bounded. Further, we have due to (3.16) $$\| {\it{\Psi}}_h^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}} \|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^{k+1}$$, which proves the estimate for the $$r=0$$ term in the sum. For the $$r=1, \ldots,k+1$$ terms, we note \begin{align*} h^r \Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} & \leq h^r \Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - I_k {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \\ & \lesssim \Vert {{\it{\Theta}}_h^{\it{\Gamma}}} - I_k {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \\ & \lesssim \Vert {{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + \Vert {\it{\Psi}}^{\it{\Gamma}} - I_k {\it{\Psi}}^{\it{\Gamma}} \Vert_{\infty,T} + h^r \Vert D^r (I_k {\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T}, \end{align*} with estimates that are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. The bound for the first term on the right-hand side follows from the previous result. The other terms are also uniformly bounded by $$h^{k+1}$$ due to the regularity of $${\it{\Psi}}^{\it{\Gamma}}$$, cf. (3.9). □ From the result in Lemma 3.7, we obtain an optimal convergence order result for the distance between the approximate interface $${\it{\Gamma}}_h = {{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})$$ and $${\it{\Gamma}}$$. Lemma 3.8 The estimate ${\rm dist}({\it{\Gamma}}_h,{\it{\Gamma}}) \lesssim h^{k+1}$ holds. Proof. Take $$x \in {\it{\Gamma}}$$. Let $$y=x+\xi \nabla \phi(x) \in {\it{\Gamma}}_h={{\it{\Theta}}_h^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}})$$ be the point closest to $$x$$. Using (2.2) and $$\|\nabla \phi(x)\|_2 \sim 1$$, we get, using $$\phi({{\it{\Psi}}^{\it{\Gamma}}}({{\it{\Gamma}}^{\mathrm{lin}}}))=0$$, \begin{align*} \|x-y\|_2 & \sim |\xi| \sim |\phi(x+\xi \nabla \phi(x)) - \phi(x)|= |\phi(x+\xi \nabla \phi(x))| \leq \|\phi\|_{\infty,{\it{\Gamma}}_h} \\ & = \|\phi \circ{{\it{\Theta}}_h^{\it{\Gamma}}}\|_{\infty, {{\it{\Gamma}}^{\mathrm{lin}}}} = \|\phi\circ{{\it{\Theta}}_h^{\it{\Gamma}}}- \phi\circ{{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty, {{\it{\Gamma}}^{\mathrm{lin}}}} \lesssim \|{{\it{\Theta}}_h^{\it{\Gamma}}} -{{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty, {{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^{k+1}. \end{align*} □ 3.3 Extension procedure In this section, we explain and analyse a general extension procedure for extending piecewise smooth functions given on $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ to the domain $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$, in such a way that the extension is zero on $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$ and piecewise smooth on $${{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$. We use a local extension procedure, introduced and analysed in Lenoir (1986); Bernardi (1989), which is a standard tool in isoparametric FEMs. In Sections 3.4 and 3.5, this procedure is applied to construct extensions of $${\it{\Psi}}^{\it{\Gamma}}$$ and $${{\it{\Theta}}_h^{\it{\Gamma}}}$$. We first describe an extension operator, introduced in Lenoir (1986), which handles the extension of a function from (only) one edge or face to a triangle or tetrahedron. The presentation here is simpler as in Lenoir (1986), because we restrict to the case $$d=2,3$$ (although this is not essential) and we only treat extension of functions given on the piecewise linear boundary $$\partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$, hence we do not have to consider the issue of boundary parametrizations. Let $$F$$ be an edge or a face of $$T\in {{\mathcal T}^{{\it{\Gamma}}}_+}\setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ from which we want to extend a function $$w \in C(F)$$, which is zero at the vertices of $$F$$, to the interior of $$T$$. $${F}$$ is called a $$1$$-face of $$T$$ if $$F$$ is an edge of the triangle $$T$$ ($$d=2$$) or an edge the tetrahedron $${T}$$ ($$d=3$$), and it is called a $$2$$-face if $$F$$ is a face of the tetrahedron $${T}$$. Applying affine linear transformations $$\hat {\it{\Phi}}_F: \hat{F} \rightarrow F$$ and $$\hat {\it{\Phi}}_T: \hat{T} \rightarrow T$$, we consider the extension problem in the reference configuration, with $$\hat{T}$$ and $$\hat{F}$$ the reference element and a corresponding face (or edge) and $$\hat{w} := w \circ \hat{\it{\Phi}}_F^{-1}$$. By $$\lambda_i, i=1,..,d+1$$, we denote the barycentric coordinates of $$\hat{T}$$, and we assume that the vertices corresponding to the coordinates $$\lambda_1, \ldots, \lambda_{p+1}$$ with coordinates $$a_1,...,a_{p+1} \in {\mathbb{R}}^d$$ are also the vertices of the $$p$$-face $$\hat{F}$$. The linear scalar weight function $$\omega$$ and the vector function $$Z$$, mapping from $$\hat{T}$$ to $$\hat{F}$$, are defined by $$\label{defomega} \omega := \sum_{i=1}^{p+1} \lambda_i \quad \text{and} \quad Z := \left(\sum_{i=1}^{p+1} \lambda_i a_i \right) / \omega.$$ (3.18) Furthermore, given the interpolation operator $${\it{\Lambda}}_l: C(\hat{F}) \rightarrow \mathcal{P}^l(\hat{F})$$, we define $$\label{defA} A_l^\ast := {\rm id} - {\it{\Lambda}}_l, \quad A_l := {\it{\Lambda}}_l - {\it{\Lambda}}_{l-1} = - A_{l}^\ast + A_{l-1}^\ast.$$ (3.19) The interpolation operator $${\it{\Lambda}}_l$$ in Lenoir (1986) is the usual nodal interpolation operator. We note, however, that this choice is not crucial. Given these components, we define the following extension operator from Lenoir (1986); Bernardi (1989): $$\label{eq:lenoirextref} \mathcal{E}^{\hat{F}\rightarrow \hat{T}} \hat w :=\omega^{k+1} A_k^\ast(\hat w)\circ Z+ \sum_{l=2}^k \omega^l A_l(\hat{w}) \circ Z, \quad \hat{w} \in C(\hat{F}).$$ (3.20) The extension in physical coordinates is then given by $$\label{eq:lenoirext} \mathcal{E}^{F \rightarrow T} w := (\mathcal{E}^{\hat{F}\rightarrow \hat{T}} (w \circ \hat{{\it{\Phi}}}_F^{-1})) \circ \hat{{\it{\Phi}}}_T^{-1}.$$ (3.21) A few elementary properties of this extension operator are collected in the following lemma. These results easily follow from Remarks 6.4–6.6 in Bernardi (1989). We use the notation $$C_0^m(F):= \{\, v \in C^m(F) | v|_{\partial F}=0\}$$. We write $$C_0(F):=C_0^0(F)$$. Lemma 3.9 The following holds: \begin{align} (\mathcal{E}^{F \rightarrow T} w)_{|F} & = w_{|F} & \text{for all} \,\, w \in C_0(F),\\ \end{align} (3.22) \begin{align} \mathcal{E}^{F \rightarrow T} w & \in \mathcal{P}^k(T)& \text{for all} \,\, w\in \mathcal{P}^k(F), \label{proppol} \\ \end{align} (3.23) \begin{align} \mathcal{E}^{F \rightarrow T} w & = 0 & \text{for all} \,\, w\in \mathcal{P}^1(F). \label{proppone} \\ \end{align} (3.24) For all $$p$$-faces $$F, \tilde F$$ of $$T$$ with $$\tilde F \neq F$$: \begin{align} (\mathcal{E}^{F \rightarrow T}w)_{|\tilde F} & = 0 & \text{for all $w \in C_0(F)$}. \label{proponeface} \\ \end{align} (3.25) Let $$T,\tilde T$$ be two tetrahedra with a common face $$F_2= T \cap \tilde T$$ and $$F_1$$ an edge of $$F_2$$. Then: \begin{align} (\mathcal{E}^{F_1 \rightarrow T}w)_{|F_2} & = (\mathcal{E}^{F_1 \rightarrow \tilde T}w)_{|F_2} & \text{for all $w \in C_0(F_1)$}. \label{propconsist} \end{align} (3.26) The following result is a key property of this extension operator. Similar results are derived in Lenoir (1986); Bernardi (1989). Lemma 3.10 For any $$p$$-face $$F$$ of $$T$$ the following holds: $$\Vert D^n \mathcal{E}^{F\rightarrow T} w \Vert_{\infty,T} \lesssim \sum_{r=n}^{k+1} h^{r-n} \Vert D^r w \Vert_{\infty,F}, \quad \forall w \in C_0^{k+1}(F), n=0,..,k+1.$$ Proof. The proof is along the same lines as the proof of Theorem 6.2 in Bernardi (1989), and is given in the Appendix A, Section A.3. □ We now describe how to obtain a ‘global’ extension $$w^{ext}$$ of a function $$w \in C(\partial{{\it{\Omega}}^{{\it{\Gamma}}}})$$. The structure of this extension is as follows. First, we introduce a simple variant of the previous extension operator $$\mathcal{E}^{F\rightarrow T}$$ to extend values from vertices and obtain $$w_0^{ext}$$. The difference $$w - w_0^{ext}$$ has, by construction, zero values on all vertices, so that we can apply the extension from edges to elements, using $$\mathcal{E}^{F\rightarrow T}$$ on each element, to obtain $$w_1^{ext}$$. In two dimensions, this already concludes the extension and we set $$w^{ext} = w_1^{ext}$$. In three dimensions we finally apply the extension from faces to elements, based on $$w - w_1^{ext}$$, which has zero values on all edges, and obtain the extended function $$w^{ext}$$. 3.3.1 The extension from vertices We define the linear interpolation $$I_1^{ext}:\, C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}}) \to V_h^1({{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$ as $$\left\{\begin{array}{@{}ll@{}} \displaystyle(I_1^{ext}w)(x_i) = w(x_i) & \text{for all vertices}\,\, x_i \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}} \\ \displaystyle(I_1^{ext}w)(x_i) = 0 & \text{at all other vertices in} \,\,{{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}, \end{array}\right.$$ (3.27) and for $$w \in C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ we set $$w_0^{ext} := I_1^{ext} w$$. With this extension, we have $$(w - w_0^{ext})(x_i) = 0$$ for all vertices $$x_i \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. On all elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ that have no edge in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$, we set $$w^{ext} = w_0^{ext}$$. All other elements are treated below. 3.3.2 The extension from edges Let $$S_1$$ be the set of edges in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal T}(F)$$ the set of elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ which have $$F$$ as an edge. We define the extension from edges as $$\label{eq:extfromedge} w_1^{ext} := w_0^{ext} + \sum_{F \in S_1} \sum_{T\in{\mathcal T}(F)} \mathcal{E}^{F\rightarrow T} (w - w_0^{ext}).$$ (3.28) We note that $$w_1^{ext}$$ is continuous due to the consistency property (3.26). On all elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ that have no face in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$, i.e., all elements in the two-dimensional case, we set $$w^{ext} := w_1^{ext}$$. All other elements are treated below with an additional extension from faces to elements. 3.3.3 The extension from faces Let $$S_2$$ be the set of faces in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal T}(F)$$ the set of elements $$T \in {{\mathcal T}^{{\it{\Gamma}}}_+} \setminus {{\mathcal T}^{{\it{\Gamma}}}}$$ which have $$F$$ as a face. Analogously to (3.28), we define the extension from faces as $$w_2^{ext} := w_1^{ext} + \sum_{F \in S_2} \sum_{T\in{\mathcal T}(F)} \mathcal{E}^{F\rightarrow T} \left(w - w_1^{ext}\right)\!.$$ (3.29) With this extension, we finally have $$(w - w_2^{ext})|_{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}} = 0$$ and set $$w^{ext} = w_2^{ext}$$. From the properties of the extension operator listed above, it follows that $$w^{ext} \in C({{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$, $$w^{ext}$$ is a continuous extension of $$w$$ and $$w=0$$ on $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$. This defines the linear extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}:\, C(\partial{{\it{\Omega}}^{{\it{\Gamma}}}}) \to C({{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus {{\it{\Omega}}^{{\it{\Gamma}}}})$$, by $$w^{ext}= \mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w$$. This operator is suitable for the extensions of $${{\it{\Theta}}_h^{\it{\Gamma}}} - {\rm id}$$ and $${{\it{\Psi}}^{\it{\Gamma}}}- {\rm id}$$, and inherits the boundedness property given in Lemma 3.10, which leads to the following result. Theorem 3.11 Let $${\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ denote the set of vertices in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $${\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ the set of all edges ($$d=2$$) or faces ($$d=3$$) in $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. The following estimates hold \begin{align} \left\|D^n\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w\right\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} & \lesssim \max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})} \sum_{r=n}^{k+1} h^{r-n} \|D^rw\|_{\infty,F} \nonumber \\ & \quad +\, h^{-n}\max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|w(x_i)|,\quad n=0,1, \label{estext1} \\ \end{align} (3.30) \begin{align} \max_{T \in {{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \left\|D^n\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}} w\right\|_{\infty,T} & \lesssim \sum_{r=n}^{k+1} h^{r-n} \|D^rw\|_{\infty,F}, \quad n=2,\ldots, k+1,\label{estext2} \end{align} (3.31) for all $$w \in C(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$ such that $$w \in C^{k+1}(F)$$ for all $$F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})$$. Proof. We note that due to (3.24) we have that $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}} = I_1^{ext} + \mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$. Furthermore $$(w - I_1^{ext}w)(x_i)=0$$ for all vertices $$x_i \in \partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$ and $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$ is a composition of the element-local extension operator $$\mathcal{E}^{F \rightarrow T}$$ only. Hence, the results in Lemma 3.10 imply corresponding results for $$\mathcal{E}^{\partial {{\it{\Omega}}^{{\it{\Gamma}}}}}({\rm id} - I_1^{ext})$$. For $$r \geq 2$$, we have $$D^rI_1^{ext}w=0$$, and furthermore $$\label{pll} \|D^r(w-I_1^{ext}w)\|_{\infty,F} \lesssim \|D^rw\|_{\infty,F}, \quad \text{for} \,\, 0 \leq r \leq k+1, \,\, \text{and} \,\, F\in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}}).$$ (3.32) From this estimate and the result in Lemma 3.10, the estimate (3.31) follows. Using $\| I_1^{ext}w\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}}+h \|DI_1^{ext}w\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|w(x_i)|$ in combination with the result in Lemma 3.10 and (3.32) yields the estimate (3.30). □ 3.4 The global mesh transformation $${\it{\Psi}}$$ Given the local transformation $${{\it{\Psi}}^{\it{\Gamma}}}$$ and the extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}$$ defined in Section 3.3, we define the following global continuous mapping on $${\it{\Omega}}$$: $$\label{eq:globext} \mathcal{E} {\it{\Psi}}^{\it{\Gamma}} = \left\{\begin{array}{@{}rcl@{}} {{\it{\Psi}}^{\it{\Gamma}}} & \text{on} & {{\it{\Omega}}^{{\it{\Gamma}}}}, \\ {\rm id} +\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id}) & \text{on} & {{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}, \\ {\rm id} & \text{on} & {\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}, \end{array}\right.$$ (3.33) and $${\it{\Psi}}:=\mathcal{E} {\it{\Psi}}^{\it{\Gamma}}$$. For this global mapping, the following bounds on derivatives are easily derived from the results already obtained for $${\it{\Psi}}^{\it{\Gamma}}$$ and for the extension operator $$\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}$$. Theorem 3.12 For $${\it{\Psi}}=\mathcal{E} {\it{\Psi}}^{\it{\Gamma}}$$, the following holds: \begin{align} \|{\it{\Psi}} - {\rm id}\|_{\infty,{\it{\Omega}}}+ h\|D{\it{\Psi}} - I\|_{\infty,{\it{\Omega}}} & \lesssim h^2, \label{bpsi1} \\ \end{align} (3.34) \begin{align} \max_{T\in{\mathcal T}} \Vert D^l {\it{\Psi}} \Vert_{\infty,T} & \lesssim 1, \quad 0 \leq l \leq k+1. \label{bpsi2} \end{align} (3.35) Proof. On $${\it{\Omega}} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}_+}$$, these results are trivial, because $${\it{\Psi}} - I=0$$. On $${{\it{\Omega}}^{{\it{\Gamma}}}}$$, we have $$D{\it{\Psi}} - I= D{{\it{\Psi}}^{\it{\Gamma}}}- I$$, and the results are a direct consequence of Corollary 3.2. On $${{\it{\Omega}}^{{\it{\Gamma}}}_+} \setminus {{\it{\Omega}}^{{\it{\Gamma}}}}$$, the result in (3.35) for $$l=0,1$$ follows from (3.34), and for $$l \geq 2$$, it follows from (3.31) combined with (3.9). We use the estimate in (3.30) and thus get \begin{align*} & \|{\it{\Psi}} - {\rm id} \|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} + h\|D{\it{\Psi}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ &\quad =\|\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}}+ h \|D\mathcal{E}^{\partial{{\it{\Omega}}^{{\it{\Gamma}}}}}({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}_+}\setminus{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ &\quad \lesssim \sum_{r=0}^{k+1} h^{r} \max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}\|D^r ({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})\|_{\infty,F}+ \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|({{\it{\Psi}}^{\it{\Gamma}}}-{\rm id})(x_i)| \\ &\quad \lesssim \|{{\it{\Psi}}^{\it{\Gamma}}} - {\rm id}\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}}+h \|D{{\it{\Psi}}^{\it{\Gamma}}} - I\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + h^2 \max_{2 \leq l \leq k+1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|D^l {{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty,T} + \max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|d(x_i)|, \end{align*} and using (3.2a) and the results of Corollary 3.2 completes the proof. □ From (3.34), it follows that, for $$h$$ sufficiently small, $${\it{\Psi}}$$ is a bijection on $${\it{\Omega}}$$. Furthermore, this mapping induces a family of (curved) finite elements that is regular of order $$k$$, in the sense as defined in Bernardi (1989). The corresponding curved finite element space is given by $$\label{FEcurvedPsi} V_{h,{\it{\Psi}}}:= \{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^k\, \}.$$ (3.36) Due to the results in Theorem 3.12, the analysis of the approximation error for this finite element space as developed in Bernardi (1989) can be applied. Corollary 4.1 from that article yields that there exists an interpolation operator $${\it{\Pi}}_h:\, H^{k+1}({\it{\Omega}}) \to V_{h,{\it{\Psi}}}$$, such that $$\label{errorBernardi} \|u- {\it{\Pi}}_h u\|_{L^2({\it{\Omega}})}+ h\|u-{\it{\Pi}}_h u\|_{H^1({\it{\Omega}})} \lesssim h^{k+1} \|u\|_{H^{k+1}({\it{\Omega}})} \quad \text{for all} u \in H^{k+1}({\it{\Omega}}).$$ (3.37) This interpolation result will be used in the error analysis of our method in Section 5.5. 3.5 The global mesh transformation $${{\it{\Theta}}_h}$$ We define the extension of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ using the same approach as for $${\it{\Psi}}^{\it{\Gamma}}$$ in (3.33), i.e., $${{\it{\Theta}}_h}:=\mathcal{E}{{\it{\Theta}}_h^{\it{\Gamma}}}$$. This global mapping $${{\it{\Theta}}_h}$$ is used in the isoparametric unfitted FEM explained in the next section. Note that $${\it{\Psi}}:=\mathcal{E}{\it{\Psi}}^{\it{\Gamma}}$$ is only used in the analysis. Remark 3.13 The mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$ is a piecewise polynomial function of degree $$k$$ on $$\partial{{\it{\Omega}}^{{\it{\Gamma}}}}$$ and thus $${{\it{\Theta}}_h} \in V_h^k({\it{\Omega}})^d$$, cf. (3.23). Further, we have $${{\it{\Theta}}_h^{\it{\Gamma}}}(x_i) = 0$$ on all vertices of $$\partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. Both lead to simplifications in the extension of $${{\it{\Theta}}_h^{\it{\Gamma}}}$$. In (3.20), the first term involving $$A_k^{\ast}$$ vanishes as $${{\it{\Theta}}_h^{\it{\Gamma}}}|_F \in \mathcal{P}^k(F)$$ for every edge and face $$F \in \partial {{\it{\Omega}}^{{\it{\Gamma}}}}$$. Furthermore, $$I_1^{ext} {{\it{\Theta}}_h^{\it{\Gamma}}} = 0$$, so that in an implementation only the extensions from edges and faces (with $$w_0^{ext} = 0$$) have to be considered. We do not address further implementation aspects of the mapping $${{\it{\Theta}}_h}$$ here. For the local mapping $${{\it{\Theta}}_h^{\it{\Gamma}}}$$, these are discussed in Lehrenfeld (2016a). The extension procedure is the same as in the well-established isoparametric FEM for a high-order boundary approximation. Here, we only note that in our implementation of this extension, we use a convenient approach based on a hierarchical/modal basis for the finite element space $$V_h^k$$ (cf. Karniadakis & Sherwin, 2013). We discuss shape regularity of the mapping $${\it{\Theta}}_h$$. Clearly, the mapping $${\it{\Theta}}_h$$ should be a bijection on $${\it{\Omega}}$$, and the transformed simplices $${\it{\Theta}}_h(T)$$, $$T \in {\mathcal T}$$, should have some shape regularity property. It is convenient to relate the transformed simplices $${\it{\Theta}}_h(T)$$ to (piecewise) transformations of the unit simplex, denoted by $$\hat T$$. The simplicial triangulation $${\mathcal T}$$ can be represented by affine transformations $${\it{\Phi}}_T(x)=A_Tx +b_T$$, i.e., $${\mathcal T}=\{\, {\it{\Phi}}_T(\hat T)\,\}$$. This defines a mapping $$\hat T \to \tilde{T}:={\it{\Theta}}_h ({\it{\Phi}}_T(\hat T))$$. This mapping should be bijective and well conditioned. Note that $$\kappa(D({{\it{\Theta}}_h} \circ {\it{\Phi}}_T)) \leq \kappa(D {{\it{\Theta}}_h}) \kappa(D {\it{\Phi}}_T)$$, where $$\kappa(\cdot)$$ denotes the spectral condition number. Because $${\it{\Phi}}_T: \hat T \to T$$ is bijective and well conditioned, it suffices to show the bijectivity and well-conditioning of $${\it{\Theta}}_h: T \to {\it{\Theta}}_h(T)$$. Note that $${\it{\Psi}}- {{\it{\Theta}}_h}= \mathcal{E}({{\it{\Psi}}^{\it{\Gamma}}}-{{\it{\Theta}}_h^{\it{\Gamma}}})$$ and using the estimate (3.30), with $$n=1$$, combined with the result in Lemma 3.7, we get $$\|D({\it{\Psi}}- {{\it{\Theta}}_h})\|_{\infty,{\it{\Omega}}} \lesssim h$$. Thus, combined with the estimate in (3.34) we get $$\label{estD} \|D{{\it{\Theta}}_h} - I\|_{\infty,{\it{\Omega}}} \lesssim h.$$ (3.38) This implies that for $$h$$ sufficiently small (the ‘resolved’ case) $$D{\it{\Theta}}_h$$ is invertible, and thus $${\it{\Theta}}_h: T \to {\it{\Theta}}_h(T)= \tilde T$$ is a bijection and furthermore $$\kappa(D {\it{\Theta}}_h) = 1+ \mathcal{O}(h)$$. Hence, we have shape regularity of $$\tilde{{\mathcal T}} = \{\tilde T\}$$ for $$h$$ sufficiently small. In the analysis, we consider only the resolved case with $$h$$ sufficiently small. In practice, one needs suitable modifications of the method to guarantee shape regularity also in cases where $$h$$ is not ‘sufficiently small’. We do not discuss this here and instead refer to Lehrenfeld (2016a). Further important consequences of the smallness of the deformation $${{\it{\Theta}}_h}$$ are summarized in the following lemma. Lemma 3.14 Let $$F := D {{\it{\Theta}}_h}$$, $$J_V := \det(F)$$, $$J_{{\it{\Gamma}}} := J_V \Vert F^{-T} n_{{\it{\Gamma}}^{\rm lin}} \Vert_2$$. There holds \begin{align} \max_{T \in {\mathcal T}} \| F - I \|_{\infty,T} & \lesssim h, \label{estF}\\ \end{align} (3.39a) \begin{align} \int_{{\it{\Omega}}_i^{\rm lin}} v^2 \, {\rm d}\tilde x \sim \int_{{\it{\Omega}}_i^{\rm lin}} J_V v^2 \, {\rm d}\tilde x &= \int_{{{\it{\Theta}}_h}({\it{\Omega}}_i^{\rm lin})} \!\!\!\!\!\!\!\! (v \circ {{\it{\Theta}}_h}^{-1})^2 \, {\rm d} x, \label{jacV}\\ \end{align} (3.39b) \begin{align} \label{jacG} \int_{{\it{\Gamma}}^{\rm lin}} v^2 \, {\rm d}\tilde s \sim \int_{{\it{\Gamma}}^{\rm lin}} J_{\it{\Gamma}} v^2 \, {\rm d}\tilde s &= \int_{{{\it{\Theta}}_h}({\it{\Gamma}}^{\rm lin})} \!\!\!\!\!\!\!\! (v \circ {{\it{\Theta}}_h}^{-1})^2 \, {\rm d}s, \\ \end{align} (3.39c) \begin{align} \label{jacgrad} \Vert \nabla (v \circ {{\it{\Theta}}_h}^{-1}) \Vert_2 & \sim \Vert \nabla v \Vert_2 \text{a.e. in} {\it{\Omega}} \text{for} v \in H^1({\it{\Omega}}). \end{align} (3.39d) Proof. From (3.38), the result (3.39a) and $$\det(F)=1+\mathcal{O}(h)$$ easily follow. The latter implies the $$\sim$$ result in (3.39b). We now consider the integral transformation result in (3.39c). We only treat $$d=3$$ ($$d=2$$ is very similar). Take $$x \in {{\it{\Gamma}}^{\mathrm{lin}}} \cap T$$ and a local orthonormal system $$t_1, t_2 \in n_{{{\it{\Gamma}}^{\mathrm{lin}}}}(x)^\perp$$ at $$x$$. The function $$g(z_1,z_2):= {{\it{\Theta}}_h}(x+z_1t_1+z_2 t_2)$$, $$z_i \in \Bbb{R}$$ parametrizes $${{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}} \cap T)$$. For the change in measure, we have, with $$F=F(x)=D{{\it{\Theta}}_h}(x)$$, $$\label{changemeas} \begin{split} {\rm d}s & = \left\| \frac{\partial g}{\partial z_1} \times \frac{\partial g}{\partial z_2}\right\|_2\, {\rm d} \tilde s =\|F t_1 \times F t_2\|_2\, {\rm d}\tilde s \\ & = |\det (F)| \|F^{-T}(t_1 \times t_2)\|_2\, {\rm d}\tilde s = \det (F) \|F^{-T}n_{{{\it{\Gamma}}^{\mathrm{lin}}}}(x)\|_2 \, {\rm d}\tilde s = J_{\it{\Gamma}} \, {\rm d}\tilde s, \end{split}$$ (3.40) which proves the equality in (3.39c). From (3.39a) and $$\det(F)=1+\mathcal{O}(h)$$, we get the $$\sim$$ result in (3.39c). With $$\nabla (v \circ {{\it{\Theta}}_h}^{-1})= F^{-T} \nabla v$$ and (3.39a), the result in (3.39d) follows. □ 4. Isoparametric unfitted FEM In this section, we introduce the isoparametric unfitted FEM based on the isoparametric mapping $${\it{\Theta}}_h$$. We consider the model elliptic interface problem (1.1). The weak formulation of this problem is as follows: determine $$u \in H_0^1({\it{\Omega}})$$ such that $\int_{{\it{\Omega}}} \alpha \nabla u \cdot \nabla v \, {\rm d}x = \int_{\it{\Omega}} f v \, {\rm d}x \quad \text{for all} \,\, v \in H_0^1({\it{\Omega}}).$ We define the isoparametric Nitsche unfitted FEM as a transformed version of the original Nitsche unfitted FE discretization (Hansbo & Hansbo, 2002) with respect to the interface approximation $${\it{\Gamma}}_h = {\it{\Theta}}_h({{\it{\Gamma}}^{\mathrm{lin}}})$$. We introduce some further notation. The standard unfitted space w.r.t. $${{\it{\Gamma}}^{\mathrm{lin}}}$$ is denoted by $V_h^{\it{\Gamma}}:= {V_h^k}_{|{{\it{\Omega}}^{\mathrm{lin}}_1}}\oplus {V_h^k}_{|{{\it{\Omega}}^{\mathrm{lin}}_2}}.$ To simplify the notation, we do not explicitly express the polynomial degree $$k$$ in $$V_h^{\it{\Gamma}}$$. Remark 4.1 Note that the polynomial degree $$k$$ is used in the FE space that contains the level set function approximation, $$\phi_h \in V_h^k$$. In the definition of the unfitted FE space $$V_h^{\it{\Gamma}}$$ above, which is used for the discretization of the interface problem (1.1), we could also use a polynomial degree $$m \neq k$$. We restrict to the case that both spaces (for the discrete level set function and for the discretization of the PDE) use the same degree $$k$$, because this simplifies the presentation and there is no significant improvement of the method if one allows $$m \neq k$$. The isoparametric unfitted FE space is defined as $$\label{transfspace} V_{h,{\it{\Theta}}}^{\it{\Gamma}}:= \{\, v_h \circ {\it{\Theta}}_h^{-1} | v_h \in V_h^{\it{\Gamma}}\, \}= \{\, \tilde v_h | \tilde v_h \circ {\it{\Theta}}_h \in V_h^{\it{\Gamma}}\, \}.$$ (4.1) Based on this space, we formulate a discretization of (1.1) using the Nitsche technique (Hansbo & Hansbo, 2002) with $${\it{\Gamma}}_h = {\it{\Theta}}_h({{\it{\Gamma}}^{\mathrm{lin}}})$$ and $${\it{\Omega}}_{i,h} = {\it{\Theta}}_h({{\it{\Omega}}^{\mathrm{lin}}_i})$$ as numerical approximation of the geometries: determine $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ such that $$\label{Nitsche1} A_h(u_h,v_h) := a_h(u_h,v_h) + N_h(u_h,v_h) = f_h(v_h) \quad \text{for all} v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}},$$ (4.2) with the bilinear forms \begin{align} a_h(u,v) & := \sum_{i=1}^2 \alpha_i \int_{{\it{\Omega}}_{i,h}} \nabla u \cdot \nabla v\, {\rm d}x,\\ \end{align} (4.3a) \begin{align} N_h(u,v) & := N_h^c(u,v) + N_h^c(v,u) + N_h^s(u,v),\\ \end{align} (4.3b) \begin{align} N_h^c(u,v) & := \int_{{\it{\Gamma}}_h} \{\!\!\{{-\alpha \nabla v}\}\!\!\} \cdot n [\hspace{-2pt}[{u}]\hspace{-2pt}]\, {\rm d}s, \quad N_h^s(u,v) := \bar \alpha \frac{\lambda}{h} \int_{{\it{\Gamma}}_h} [\hspace{-2pt}[{u}]\hspace{-2pt}] [\hspace{-2pt}[{v}]\hspace{-2pt}]\, {\rm d}s \end{align} (4.3c) for $$u, v \in V_{h,{\it{\Theta}}}^{\it{\Gamma}} + {V_{\text{reg},h}}$$ with $${V_{\text{reg},h}} := H^1({\it{\Omega}}) \cap H^2({\it{\Omega}}_{1,h} \cup {\it{\Omega}}_{1,h})$$. Here, $$n = n_{{\it{\Gamma}}_h}$$ denotes the outer normal of $${\it{\Omega}}_{1,h}$$ and $$\bar \alpha = \frac12(\alpha_1 + \alpha_2)$$ the mean diffusion coefficient. For the averaging operator $$\{\!\!\{{\cdot}\}\!\!\}$$, there are different possibilities. We use $$\{\!\!\{w \}\!\!\} := \kappa_1 w_{|{\it{\Omega}}_{1,h}} + \kappa_2w_{|{\it{\Omega}}_{2,h}}$$ with a ‘Heaviside’ choice where $$\kappa_i = 1$$ if $$|T_i| > \frac12 |T|$$ and $$\kappa_i = 0$$ if $$|T_i| \leq \frac12 |T|$$. Here, $$T_i = T \cap {{\it{\Omega}}^{\mathrm{lin}}_i}$$, i.e., the cut configuration on the undeformed mesh is used. This choice in the averaging renders the scheme in (4.2) stable (for sufficiently large $$\lambda$$) for arbitrary polynomial degrees $$k$$, independent of the cut position of $${\it{\Gamma}}$$, cf. Lemma 5.4 below. A different choice for the averaging which also results in a stable scheme is $$\kappa_i = |T_i|/|T|$$. To define the right-hand side functional $$f_h$$, we first assume that the source term $$f_i :\, {\it{\Omega}}_i \to \Bbb{R}$$ in (1.1a) is (smoothly) extended to $${\it{\Omega}}_{i,h}$$, such that $$f_i= f_{i,h}$$ on $${\it{\Omega}}_i$$ holds. This extension is denoted by $$f_{i,h}$$. We define $$\label{eq:lfs} f_h(v) := \sum_{i=1,2} \int_{{\it{\Omega}}_{i,h}} f_{i,h} v\, {\rm d}x.$$ (4.4) We define $$f_h$$ on $${\it{\Omega}}$$ by $${f_h}_{|{\it{\Omega}}_{i,h}}:= f_{i,h}$$, $$i=1,2$$. For the implementation of this method, in the integrals, we apply a transformation of variables $$y:={\it{\Theta}}_h^{-1}(x)$$. This results in the following representations of the bi- and linear forms: \begin{align} a_h(u,v) & = \sum_{i=1,2} \alpha_i \int_{{{\it{\Omega}}^{\mathrm{lin}}_i}} D{\it{\Theta}}_h^{-T} \nabla u \cdot D{\it{\Theta}}_h^{-T} \nabla v \det (D{\it{\Theta}}_h)\, {\rm d}y, \\ \end{align} (4.5a) \begin{align} f_h(v) & = \sum_{i=1,2} \int_{{{\it{\Omega}}^{\mathrm{lin}}_i}} (f_{i,h} \circ {\it{\Theta}}_h) v \det (D{\it{\Theta}}_h)\, {\rm d}y, \\ \end{align} (4.5b) \begin{align} N_h^c(u,v) & = \int_{{{\it{\Gamma}}^{\mathrm{lin}}}} \det(D {\it{\Theta}}_h) D{\it{\Theta}}_h^{-T} \{\!\!\{{-\alpha \nabla v}\}\!\!\} \cdot (D{\it{\Theta}}_h^{-T} \cdot n^{\text{lin}}) [\hspace{-2pt}[{u}]\hspace{-2pt}]\, {\rm d}y, \label{q} \\ \end{align} (4.5c) \begin{align} N_h^s(u,v) & = \bar \alpha \frac{\lambda}{h} \int_{{{\it{\Gamma}}^{\mathrm{lin}}}} \mathcal{J}_{\it{\Gamma}} [\hspace{-2pt}[{u}]\hspace{-2pt}] [\hspace{-2pt}[{v}]\hspace{-2pt}]\, {\rm d}y, \end{align} (4.5d) where $$\mathcal{J}_{{\it{\Gamma}}} = \det (D{\it{\Theta}}_h) \Vert D {\it{\Theta}}_h^{-T} \cdot n^{\text{lin}} \Vert$$ is the ratio between the measures on $${\it{\Gamma}}_h$$ and $${{\it{\Gamma}}^{\mathrm{lin}}}$$, and $$n^{\text{lin}} = \nabla {\hat \phi_h} / \Vert \nabla {\hat \phi_h} \Vert$$ is the normal to $${{\it{\Gamma}}^{\mathrm{lin}}}$$. We note that in (4.5c) we exploited that the normalization factor $$\Vert D {\it{\Theta}}_h^{-1} \cdot n^{\text{lin}} \Vert$$ for the normal direction $$n$$ of $${\it{\Gamma}}_h$$ cancels out with the corresponding term in $$\mathcal{J}_{{\it{\Gamma}}}$$. Based on this transformation, the implementation of integrals is carried out as for the case of the piecewise planar interface $${{\it{\Gamma}}^{\mathrm{lin}}}$$. The additional variable coefficients $$D {{\it{\Theta}}_h}^{-T}$$, $$\det(D {{\it{\Theta}}_h})$$ are easily and efficiently computable using the property that $${{\it{\Theta}}_h}$$ is a finite element (vector) function. The integrands in (4.5) are in general not polynomial, so that exact integration can typically not be guaranteed. For a discussion of the thereby introduced additional consistency error, we refer to Remark 5.9 in the analysis. 4.1 Numerical experiment In this section, we present results of a numerical experiment for the method (4.2). The domain is $${\it{\Omega}} = [-1.5,1.5]^2$$, and the interface is given as $${\it{\Gamma}} = \{\phi(x) = 0 \}$$ with the level set function $$\phi(x) = \Vert x \Vert_4 - 1$$. Here, $$\Vert x \Vert_4 := (\sum_{i=1}^d x_i^4)^{\frac14}$$. The zero level of $$\phi$$ describes a ‘smoothed square’ and $$\phi$$ is equivalent to a signed distance function in the sense that $$\sqrt[4]{1/2} \leq \Vert \nabla \phi \Vert_2 \leq 1$$, such that $$\mathrm{dist}(x,{\it{\Gamma}}) \leq \sqrt[4]{2} \vert \phi(x) \vert\text{for all}x\in{\it{\Omega}}$$. The level set function $$\phi$$ is approximated with $$\phi_h \in V_h^k$$ by interpolation. For the problem in (1.1), we take the diffusion coefficient $$(\alpha_1,\alpha_2) = (1,2)$$ and the Dirichlet boundary conditions and right-hand side $$f$$, such that the solution is given by $$\label{eq:solution} u(x) = \left\{\begin{array}{@{}rc@{}}\displaystyle 1 + \frac{\pi}{2} - \sqrt{2} \cdot \cos\left(\frac{\pi}{4} \Vert x \Vert_4^4\right), & \ x \in {\it{\Omega}}_1, \\ \displaystyle\frac{\pi}{2} \Vert x \Vert_4\hphantom{)}, & \ x \in {\it{\Omega}}_2. \end{array} \right.$$ (4.6) We note that $$u$$ is continuous on $${\it{\Omega}}$$, but has a kink across the interface $${\it{\Gamma}}$$, such that $$\alpha_1 \nabla u_1 \cdot n_{\it{\Gamma}} = \alpha_2 \nabla u_2 \cdot n_{\it{\Gamma}}$$, cf. the sketch of the solution in Fig. 3. Fig. 3. View largeDownload slide Sketch of domains $${\it{\Omega}}_1$$, $${\it{\Omega}}_2$$ (left), the solution (middle and right) and the initial mesh ($$L=0$$, right) for the example in Section 4.1. Fig. 3. View largeDownload slide Sketch of domains $${\it{\Omega}}_1$$, $${\it{\Omega}}_2$$ (left), the solution (middle and right) and the initial mesh ($$L=0$$, right) for the example in Section 4.1. For the construction of the mapping $${{\it{\Theta}}_h}$$, the one-dimensional nonlinear problem that results from (3.12) is solved with a Newton iteration, cf. (Lehrenfeld, 2016a, section 2.3), up to a (relative) tolerance of $$10^{-14}$$, which required only two to three iterations for each point. For the search direction, we chose $$G_h = \nabla \phi_h$$, cf. (3.10). We also considered $$G_h=P_h^{\it{\Gamma}} \nabla \phi_h$$, which led to very similar results. For the computation of the integrals, we apply the transformation to the reference domains $${{\it{\Omega}}^{\mathrm{lin}}_i}$$ and $${{\it{\Gamma}}^{\mathrm{lin}}}$$ as in (4.5). On each element of the reference domain (e.g., $${{\it{\Omega}}^{\mathrm{lin}}_i} \cap T$$, $$T \in {\mathcal T}_h$$), we apply quadrature rules of exactness degree $$2k-2$$ for the integrals in $$a_h(\cdot,\cdot)$$ and $$N_h^c(\cdot,\cdot)$$, and quadrature rules of exactness degree $$2k$$ for the integrals in $$N_h^s(\cdot,\cdot)$$, see also remark 5.9 below. The method has been implemented in the add-on library ngsxfem to the finite element library NGSolve (Schöberl, 2014). We chose an initial simplicial mesh ($$L=0$$) with $$282$$ elements, which sufficiently resolves the interface such that the shape regularity of the mesh (after transformation) is given without a further limitation step as discussed in Lehrenfeld (2016a, Section 2.6). Starting from this initial triangulation, uniform refinements are applied. The stabilization parameter $$\lambda$$ in the unfitted Nitsche method is chosen as $$\lambda = 20 \cdot k^2$$. With the discrete solution to (4.2) $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, and $$u$$ as in (4.6), we define the following error quantities that we can evaluate for the numerical solutions: \begin{align*} d_{{\it{\Gamma}}_h}\! :=\! \Vert \phi \Vert_{\infty,{\it{\Gamma}}_h}, \quad e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}:= \Vert [\hspace{-2pt}[{E_i u - u_h}]\hspace{-2pt}] \Vert_{L^2({\it{\Gamma}}_h)} \\ e_{L^2}^2\! :=\! \sum_{i=1,2} \Vert E_i u - u_h\Vert_{L^2({\it{\Omega}}_{i,h})}^2, \quad e_{H^1}^2\! :=\! \sum_{i=1,2} \Vert \nabla(E_i u - u_h) \Vert_{L^2({\it{\Omega}}_{i,h})}^2. \end{align*} Here, $$E_i:{\it{\Omega}}_i \rightarrow {\it{\Omega}}_{i,h}={{\it{\Theta}}_h}({{\it{\Omega}}^{\mathrm{lin}}_i})$$ denotes the canonical extension operator for the solution (using the representation in (4.6)). Due to the equivalence of $$\phi$$ to a signed distance function we have $$\mathrm{dist}({\it{\Gamma}}_h,{\it{\Gamma}}) \leq \sqrt[4]{2} d_{{\it{\Gamma}}_h}$$, and thus $$d_{{\it{\Gamma}}_h}$$ yields a (sharp) bound for the error in the geometry approximation. The error quantities $$e_{H^1}$$, $$d_{{\it{\Gamma}}_h}$$ and $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ are also used in the error analysis, cf. (5.1). The results in the experiment confirm the theoretical error bounds. We also include the $$L^2$$-error $$e_{L^2}$$, although in the analysis in this article, we do not derive $$L^2$$-error bounds, cf. Section 6. Results are listed in Table 1. We observe $$d_{{\it{\Gamma}}_h} \sim \mathcal{O}(h^{k+1})$$, $$e_{L^2} \sim \mathcal{O}(h^{k+1})$$, $$e_{H^1} \sim \mathcal{O}(h^{k})$$, i.e., optimal order of convergence in these quantities. The observed convergence rate for the jump across the interface, $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]} \sim \mathcal{O}(h^{k+1})$$, is better than that predicted in the analysis below (by half an order). For a fixed mesh ($$L=const$$), we observe that increasing the polynomial degree $$k$$ dramatically decreases the error in all four error quantities. The discretization with $$k=6$$ on the coarsest mesh $$L=0$$ with only $$6835$$ unknowns (last row in the table) is much more accurate in all the four quantities than the discretization with $$k=1$$ after six additional mesh refinements and 583 401 unknowns (row $$k=1$$, $$L=6$$ in the table). Table 1 Discretization errors and estimated orders of convergence (eoc) for the example inSection 4.1 $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) Table 1 Discretization errors and estimated orders of convergence (eoc) for the example inSection 4.1 $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) $$k$$ $$L$$ $$d_{{\it{\Gamma}}_h}$$ (eoc) $$e_{L^2}$$ (eoc) $$e_{H^1}$$ (eoc) $$e_{[\hspace{-2pt}[{\cdot}]\hspace{-2pt}]}$$ (eoc) 1 0 $${5.022} \times 10^{-3}$$ (—) $${8.502} \times 10^{-2}$$ (—) $${6.872} \times 10^{-1}$$ (—) $${3.738} \times 10^{-3}$$ (—) 1 $${2.239} \times 10^{-3}$$ (1.2) $${2.454} \times 10^{-2}$$ (1.8) $${4.048} \times 10^{-1}$$ (0.8) $${1.652} \times 10^{-3}$$ (1.2) 2 $${5.390} \times 10^{-4}$$ (2.0) $${6.112} \times 10^{-3}$$ (2.0) $${2.100} \times 10^{-1}$$ (0.9) $${3.862} \times 10^{-4}$$ (2.1) 3 $${1.238} \times 10^{-4}$$ (2.1) $${1.503} \times 10^{-3}$$ (2.0) $${1.057} \times 10^{-1}$$ (1.0) $${1.138} \times 10^{-4}$$ (1.8) 4 $${3.364} \times 10^{-5}$$ (1.9) $${3.730} \times 10^{-4}$$ (2.0) $${5.342} \times 10^{-2}$$ (1.0) $${2.743}\times 10^{-5}$$ (2.1) 5 $${8.862} \times 10^{-6}$$ (1.9) $${9.350} \times 10^{-5}$$ (2.0) $${2.678} \times 10^{-2}$$ (1.0) $${6.956}\times 10^{-6}$$ (2.0) 6 $${2.259} \times 10^{-6}$$ (2.0) $${2.367} \times 10^{-5}$$ (2.0) $${1.343} \times 10^{-2}$$ (1.0) $${1.756} \times 10^{-6}$$ (2.0) 2 0 $${6.993} \times 10^{4}$$ (—) $${3.021}\times 10^{-3}$$ (—) $${1.522} \times 10^{-1}$$ (—) $${1.134} \times 10^{-4}$$ (—) 1 $${6.514} \times 10^{5}$$ (3.4) $${5.893} \times 10^{-4}$$ (2.4) $${5.152} \times 10^{-2}$$ (1.6) $${3.100}\times 10^{-5}$$ (1.9) 2 $${1.197} \times 10^{5}$$ (2.4) $${7.899} \times 10^{-5}$$ (2.9) $${1.369} \times 10^{-2}$$ (1.9) $${3.653} \times 10^{-6}$$ (3.1) 3 $${2.111} \times 10^{6}$$ (2.5) $${9.976} \times 10^{-6}$$ (3.0) $${3.446} \times 10^{-3}$$ (2.0) $${4.822} \times 10^{-7}$$ (2.9) 4 $${2.447} \times 10^{7}$$ (3.1) $${1.265} \times 10^{-6}$$ (3.0) $${8.704} \times 10^{-4}$$ (2.0) $${5.481} \times 10^{-8}$$ (3.1) 3 0 $${8.699} \times 10^{-5}$$ (—) $${1.666} \times 10^{-3}$$ (—) $${2.756} \times 10^{-2}$$ (—) $${5.289} \times 10^{-6}$$ (—) 1 $${1.960} \times 10^{-5}$$ (2.1) $${1.968} \times 10^{-4}$$ (3.1) $${3.700} \times 10^{-3}$$ (2.0) $${5.329} \times 10^{-7}$$ (3.3) 2 $$4{1.253} \times 10^{-6}$$ (4.0) $${1.179} \times 10^{-5}$$ (4.1) $${4.426} \times 10^{-4}$$ (3.1) $${2.004} \times 10^{-8}$$ (4.7) 3 $${6.392} \times 10^{-8}$$ (4.3) $${7.144} \times 10^{-7}$$ (4.0) $${5.276} \times 10^{-5}$$ (3.1) $${1.414}\times 10^{-9}$$ (3.8) 4 0 $${2.013}\times 10^{-5}$$ (—) $${9.662}\times 10^{-5}$$ (—) $${7.825} \times 10^{-3}$$ (—) $${5.744} \times 10^{-7}$$ (—) 1 $${1.028} \times 10^{-6}$$ (4.3) $${2.554} \times 10^{-6}$$ (5.2) $${6.211} \times 10^{-4}$$ (3.7) $${3.229} \times 10^{-8}$$ (4.2) 2 $${2.519} \times 10^{-8}$$ (5.4) $${9.798} \times 10^{-8}$$ (4.7) $${3.523} \times 10^{-5}$$ (4.1) $${1.066} \times 10^{-9}$$ (4.9) 5 0 $${2.473} \times 10^{-6}$$ (—) $${1.064}\times 10^{-4}$$ (—) $${1.039} \times 10^{-3}$$ (—) $${3.014} \times 10^{-8}$$ (—) 1 $${2.033} \times 10^{-7}$$ (3.6) $${1.590} \times 10^{-6}$$ (6.1) $${2.885}\times 10^{-5}$$ (5.2) $${1.803} \times 10^{-9}$$ (4.1) 6 0 $${9.328} \times 10^{-7}$$ (—) $${1.491} \times 10^{-6}$$ (—) $${2.549} \times 10^{-4}$$ (—) $${7.106} \times 10^{-9}$$ (—) Remark 4.2 We note that the arising linear systems depend on the position of the interface within the computational mesh and can become arbitrarily ill-conditioned. To circumvent the impact of this, we used a sparse direct solver for the solution of linear systems. In our experiment, the very poor conditioning of the stiffness matrix for high-order discretizations limited the achievable accuracy to $$e_{L^2} \approx 1\times 10^{-6}$$ and $$e_{H^1} \approx 1\times 10^{-6}$$. Hence, we stopped the refinement if discretization errors in this order of magnitude have been reached. The solution of linear systems is an important issue for high-order unfitted discretizations and requires further attention. To deal with the ill-conditioning, one may consider (new) preconditioning strategies or further stabilization mechanisms in the weak formulation. We leave this as a topic for future research. 5. Error analysis The main new contribution of this article is an error analysis of the method presented in Section 4. We start with the definition of norms and derive properties of the discrete variational formulation in Section 5.1. These results are obtained using the fact that the isoparametric method can be seen as a small perturbation of the standard unfitted Nitsche-XFEM method. To bound the error terms, we use the bijective mapping on $${\it{\Omega}}$$ given by $${\it{\Phi}}_h:= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$. This mapping has the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$ and is close to the identity. Some further relevant properties of this mapping are treated in Section 5.2. Using this mapping, we derive a Strang lemma-type result, which relates the discretization error to consistency errors in the bilinear form and right-hand side functional, and to approximation properties of the finite element space, cf. Section 5.3. In Sections 5.4 and 5.5, we prove bounds for the consistency and approximation error, respectively. Finally, in Section 5.6, an optimal-order $$H^1$$-error bound is given. 5.1 Norms and properties of the bilinear forms In the error analysis, we use the norm \begin{align} \Vert v \Vert_{h}^2 & := \vert v \vert_{1}^2 + \Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2 + \Vert \{\!\!\{{\alpha \nabla v}\}\!\!\} \Vert_{-\frac12,h,{\it{\Gamma}}_h}^2, \label{defn} \\ \end{align} (5.1) \begin{align} \text{with} \Vert v \Vert_{\pm \frac12,h,{\it{\Gamma}}_h}^2 & := \left(\bar{\alpha} / h \right)^{\pm 1} \Vert v \Vert_{L^2({\it{\Gamma}}_h)}^2 \text{and} |v|_1^2:= \sum_{i=1,2} \alpha_i \Vert \nabla v \Vert_{L^2({\it{\Omega}}_{i,h})}^2. \label{ppp} \end{align} (5.2) Note that the norms are formulated with respect to $${\it{\Omega}}_{i,h} = {{\it{\Theta}}_h}({{\it{\Omega}}^{\mathrm{lin}}_i})$$ and $${\it{\Gamma}}_{h} = {{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}})$$, and include a scaling depending on $$\alpha$$. Remark 5.1 For simplicity, we restrict to the setting of a quasi-uniform family of triangulations. In the more general case of a shape-regular (not necessarily quasi-uniform) family of triangulations, one has to replace the interface norm in (5.2) by one with an element wise scaling: $$\label{defr} \Vert v \Vert_{\pm \frac12,h,{\it{\Gamma}}_h}^2 := \sum_{T \in {\mathcal T}^{\it{\Gamma}}} \left(\bar{\alpha} / h_T \right)^{\pm 1} \Vert v \Vert_{L^2({\it{\Gamma}}_{h,T})}^2, \text{with} {\it{\Gamma}}_{h,T} := {{\it{\Theta}}_h}(T\cap {{\it{\Gamma}}^{\mathrm{lin}}}).$$ (5.3) We further define the space of sufficiently smooth functions that allow for the evaluation of normal gradients at the interface $${\it{\Gamma}}_h$$: $$\label{defVreg} {V_{\text{reg},h}} := H^1({\it{\Omega}}) \cap H^2({\it{\Omega}}_{1,h} \cup {\it{\Omega}}_{2,h}).$$ (5.4) Note that the norm $$\|\cdot\|_h$$ and the bilinear forms in (4.3) are well defined on $${V_{\text{reg},h}}+V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. The stability of the method relies on an appropriate choice for the weighting operator $$\{\!\!\{{u}\}\!\!\} = \kappa_1 u_1 + \kappa_2 u_2$$ in (4.2). For the stability analysis of the weighting operator, we introduce the criterion $$\label{eq:crit} \kappa_i^2 \leq c_{\kappa} |T_i|/|T|, i=1,2,$$ (5.5) with a constant $$c_{\kappa}$$ independent of $$\alpha$$, $$h$$ and the cut configuration. We note that this criterion is fulfilled for the Heaviside choice, cf. Section 4, with $$c_{\kappa}=2$$ and the weighting $$\kappa_i =|T_i|/|T|$$ proposed in Hansbo & Hansbo (2002) with $$c_{\kappa}=1$$. Using this criterion, we derive the following inverse estimate, which is an important ingredient in the stability analysis of the unfitted Nitsche method. Lemma 5.2 On a shape-regular (not necessarily quasi-uniform) family of triangulations with an averaging operator $$\{\!\!\{{u}\}\!\!\}$$ satisfying (5.5), there holds $$\label{inverse} \Vert \{\!\!\{{\alpha \nabla u}\}\!\!\} \Vert_{-\frac12,h,{\it{\Gamma}}_h} \lesssim \vert u \vert_{1} \quad \text{for all} \,\, u \in V_h^{\it{\Gamma}}.$$ (5.6) Proof. It suffices to derive the localized estimate for an arbitrary element $$T \in {\mathcal T}^{\it{\Gamma}}$$. Note that for the general case of a not necessarily quasi-uniform family of triangulations, the norm used on the left-hand side in (5.6) is as in (5.3). We only have to show $$h_T \kappa_i^2 \Vert p \Vert_{L^2({\it{\Gamma}}_{h,T})}^2 \lesssim \Vert p \Vert_{L^2({{\it{\Theta}}_h}(T_i))}^2 \quad \text{for all} \,\, p \circ {{\it{\Theta}}_h} \in \mathcal{P}^k(T),$$ with $${\it{\Gamma}}_{h,T} := {{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}_T})$$, $${{\it{\Gamma}}^{\mathrm{lin}}_T} := {{\it{\Gamma}}^{\mathrm{lin}}} \cap T$$ and $$T_i := {{\it{\Omega}}^{\mathrm{lin}}_i} \cap T$$ for $$i=1,2$$. With Lemma 3.14, this is equivalent to the corresponding result on the undeformed domains: $$h_T \kappa_i^2 \Vert p \Vert_{L^2({{\it{\Gamma}}^{\mathrm{lin}}_T})}^2 \lesssim \Vert p \Vert_{L^2(T_i)}^2 \quad \text{for all} \,\, p \in \mathcal{P}^k(T).$$ We transform the problem to the reference element $$\hat{T} := \{x \in {\mathbb{R}}^d | \sum_{i=1}^d x_i \leq 1, x_i \geq 0, i=1,..,d \}$$ with the corresponding affine linear transformation $${\it{\Phi}}_T: \hat{T} \rightarrow T$$, such that $$T = {\it{\Phi}}_T(\hat{T})$$. We introduce $$J_V := {|}\det(D {\it{\Phi}}_T)|$$, $$J_{\it{\Gamma}} := J_V \Vert (D {\it{\Phi}}_T)^{-T} n_{\hat{{\it{\Gamma}}}} \Vert_2$$, $$\hat{{\it{\Gamma}}} = {\it{\Phi}}_T^{-1} ({{\it{\Gamma}}^{\mathrm{lin}}_T})$$ and $$\hat{T}_i := {\it{\Phi}}_T^{-1} (T_i)$$ and then have $$\int_{T_i} p^2 \, {\rm d}x = J_V \int_{\hat{T}_i} (p \circ {\it{\Phi}}_T)^2 \, {\rm d}x$$, and with the same arguments as in (3.40) one gets $$\int_{{{\it{\Gamma}}^{\mathrm{lin}}_T}} p^2 \, {\rm d}s = J_{\it{\Gamma}} \int_{\hat{{\it{\Gamma}}}} (p \circ {\it{\Phi}}_T)^2 \, {\rm d}s.$$ Note that $\|D {\it{\Phi}}_T\|_2^{-1} \leq \Vert (D {\it{\Phi}}_T)^{-T} n_{\hat{{\it{\Gamma}}}} \Vert_2 \leq \|D {\it{\Phi}}_T^{-1}\|_2.$ Due to the assumption of shape regularity, we have $$\|D {\it{\Phi}}_T\|_2 \lesssim h_T$$, $$\|D {\it{\Phi}}_T^{-1}\|_2 \lesssim h_T^{-1}$$ and thus we get $$J_V \sim J_{\it{\Gamma}} h_T$$. Hence, it suffices to show $$\label{investhat} \kappa_i^2 \Vert p \Vert_{\hat{{\it{\Gamma}}}}^2 \leq c_{k,d} \Vert p \Vert_{\hat{T}_i}^2 \quad \text{for all} \,\, p \in \mathcal{P}^k(\hat{T}), i=1,2,$$ (5.7) with a constant $$c_{k,d}$$ depending only on the polynomial degree $$k$$, the dimension $$d$$ and the constant $$c_{\kappa}$$. We now prove the estimate (5.7). As $$\hat{{\it{\Gamma}}}$$ is planar $$\hat{T}_i$$ is a convex polytope. Let $$S$$ be the hyperplane, with normal denoted by $$n_S$$, which contains $$\hat{{\it{\Gamma}}}$$, and $$R = \{x \in S | x + \alpha n_S \in \hat{T}_i, \alpha \in {\mathbb{R}}\}$$ be the normal projection of $$\hat{T}_i$$ into $$S$$, cf. Fig. 4. With $${\mathcal I}_i$$ the subset of those vertices of $$\hat T$$ that are vertices of $$\hat{T}_i$$, we set $$D := \max_{x_V \in {\mathcal I}_i}\mathrm{dist}(S,x_V)$$. We define the cylinder $$T_{R,D} = \{x \in {\mathbb{R}}^d | x = y + \alpha n_S, y \in R, \alpha \in [0,D]\}$$. Note that $$\hat{T}_i \subset T_{R,D}$$ holds. We can bound the volume of $$T_{R,D}$$ by $$|T_{R,D}| \leq |R| D \leq l^{d-1} D$$, where $$l$$ is the maximum distance of two points in $$R$$, which can be bounded by the length of the longest edge of $$\hat{T}$$, i.e., $$l \leq \sqrt{2}$$. Combining this with $$|\hat{T}_i| \geq \frac{\kappa_i^2}{c_{\kappa}} |\hat{T}|$$ from (5.5), we get $$\frac{\kappa_i^2}{c_{\kappa} d!} = \frac{\kappa_i^2}{c_{\kappa}} |\hat{T}| \leq |\hat{T}_i| \leq |T_{R,D}| \leq l^{d-1} D,$$ and thus $$D \geq c_d \frac{\kappa_i^2}{c_{\kappa}}$$ with $$c_d := (d! 2^{\frac{d-1}{2}})^{-1}$$. From this, it follows that there exists a vertex $$x_V$$ of the reference simplex $$\hat T$$ such that $$x_V \in \hat{T}_i$$ and $$\mathrm{dist}(S,x_V) \geq c_d \frac{\kappa_i^2}{c_{\kappa}}$$. Note that $$\mathrm{conv}(\hat{{\it{\Gamma}}},x_V) \subset \hat{T}_i$$. Based on a decomposition of $$\hat{{\it{\Gamma}}}$$ into $$(d-1)$$-simplices $$\{G_l\}_{l=1,..,L}$$, we consider the (nonoverlapping) decomposition of $$\mathrm{conv}(\hat{{\it{\Gamma}}},x_V)$$ into $$d$$-simplices $$T_l:=\mathrm{conv}(G_l,x_V)$$, $$l=1,..,L$$. On $$T_l$$ we apply known inverse trace estimates for polynomials, c.f. (Warburton & Hesthaven, 2003): $$\Vert p \Vert_{L^2(G_l)}^2 \leq \frac{(k+d)(k+1)}{d} \frac{|G_l|}{|T_l|} \Vert p \Vert_{L^2(T_l)}^2$$. With $$|T_l| = \frac{D}{d} |G_l| \geq \frac{\kappa_i^2}{c_{\kappa}} \frac{c_d}{d} |G_l|$$, we obtain $$\kappa_i^2 | G_l | / |T_l| \leq \frac{d}{c_d} c_\kappa$$ and thus, $$\kappa_i^2 \Vert p \Vert_{L^2(G_l)}^2 \leq c_{\kappa} \frac{(k+d)(k+1)}{c_d} \Vert p \Vert_{L^2(T_l)}^2.$$ By summing over $$l=1,..,L$$, this proves (5.7) with $$c_{k,d} = c_{\kappa} d! 2^{\frac{d-1}{2}} (k+d)(k+1)$$. □ Fig. 4. View largeDownload slide Sketch of geometries used in the proof of Lemma 5.2. Fig. 4. View largeDownload slide Sketch of geometries used in the proof of Lemma 5.2. Remark 5.3 The condition (5.5) on the averaging operator is crucial for the estimate (5.6) to hold. Furthermore, the fact that the interface is piecewise planar w.r.t., the reference simplex is used in the analysis. We briefly comment on similar results from the literature. For $$k=1$$ and the weighting $$\kappa_i =|T_i|/|T|$$, the inverse estimate has been shown in Hansbo & Hansbo (2002). In Massjung (2012), the result in Lemma 5.2 has been proven for the Heaviside choice $$\kappa_i \in \{0,1 \}$$ for the case of a higher-order unfitted discontinuous Galerkin discretization for smooth interfaces in two dimensions. A variant of this method has been addressed in Wu & Xiao (2010). We are not aware of any literature in which such a result for higher-order discretizations and dimension $$d \geq 3$$ has been derived. A large part of the analysis of the usual unfitted FEMs can be carried over using the following result: Lemma 5.4 For $$\lambda$$ sufficiently large the estimates \begin{align} A_h(u,u) &\gtrsim \Vert u \Vert_h^2 && \text{for all} \, u \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}, \label{eq:coerc} \\ \end{align} (5.8) \begin{align} A_h(u,v) &\lesssim \Vert u \Vert_h \Vert v \Vert_h && \text{for all} \, u,v \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}} \label{eq:bound} \end{align} (5.9) hold. Proof. We have $$\label{r1} a_h(u,u) \sim |u|_1^2 \quad \text{for} \,\, u \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}}.$$ (5.10) Furthermore, $$\label{r2} \begin{split} |N_h^s(u,v)| & \lesssim \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} \Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}, \quad u,v \in H^1({\it{\Omega}}),\\ N_h^s(u,u) & \sim \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2, \quad u \in H^1({\it{\Omega}}). \end{split}$$ (5.11) For the Nitsche consistency term, we have, for $$u,v \in {V_{\text{reg},h}} + V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: $$\label{r3} |N_h^c(u,v)| \lesssim \Vert \{\!\!\{{\alpha \nabla u}\}\!\!\} \Vert_{-\frac12,h,{{\it{\Gamma}}^{\mathrm{lin}}}}\Vert [\hspace{-2pt}[{v}]\hspace{-2pt}] \Vert_{\frac12,h,{{\it{\Gamma}}^{\mathrm{lin}}}}.$$ (5.12) The result in (5.9) follows from the definition of $$A_h(\cdot,\cdot)$$ and (5.10), (5.11), (5.12). Using (5.6) and the results in (5.10), (5.11), (5.12), we get, for $$u \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, \begin{align*} A_h(u,u) & \gtrsim \vert u \vert_{1}^2 + \lambda \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2 - \vert u \vert_{1} \, \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} \\ & \gtrsim \vert u \vert_{1}^2 + \Vert [\hspace{-2pt}[{u}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h}^2, \end{align*} provided $$\lambda$$ is chosen sufficiently large. Using (5.6) again, we obtain the estimate (5.8). □ In the remainder, we assume that $$\lambda$$ is taken sufficiently large, such that the results in Lemma 5.4 hold. Then, the discrete problem (4.2) has a unique solution. 5.2 The bijective mapping $${\it{\Phi}}_h= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$ From the properties derived in Sections 3.4 and 3.5, it follows that, for $$h$$ sufficiently small, the mapping $${\it{\Phi}}_h= {\it{\Psi}} \circ {{\it{\Theta}}_h}^{-1}$$ is bijection on $${\it{\Omega}}$$ and has the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$. In the remainder, we assume that $$h$$ is sufficiently small such that $${\it{\Phi}}_h$$ is a bijection. It has the smoothness property $${\it{\Phi}}_h \in C({\it{\Omega}})^d \cap C^{k+1}({\mathcal T})^d$$. In the following lemma, we derive further properties of $${\it{\Phi}}_h$$ that will be needed in the error analysis. Lemma 5.5 The following holds: \begin{align} \|{{\it{\Theta}}_h} - {\it{\Psi}}\|_{\infty,{\it{\Omega}}}+ h\|D({{\it{\Theta}}_h} - {\it{\Psi}})\|_{\infty,{\it{\Omega}}} & \lesssim h^{k+1} \label{estt1} \\ \end{align} (5.13) \begin{align} \|{\it{\Phi}}_h - {\rm id}\|_{\infty,{\it{\Omega}}}+ h\|D{\it{\Phi}}_h - I\|_{\infty,{\it{\Omega}}} & \lesssim h^{k+1}.\label{estt2} \end{align} (5.14) Proof. Using the definition $${\it{\Psi}}=\mathcal{E}{{\it{\Psi}}^{\it{\Gamma}}}$$, $${{\it{\Theta}}_h}=\mathcal{E}{{\it{\Theta}}_h^{\it{\Gamma}}}$$ and the results in Theorem 3.11 and Lemma 3.7, we get the result in (5.13) with \begin{align*} &\|{{\it{\Theta}}_h} - {\it{\Psi}}\|_{\infty,{\it{\Omega}}} + h\|D({{\it{\Theta}}_h} - {\it{\Psi}})\|_{\infty,{\it{\Omega}}} \\ &\quad \lesssim \|{{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}}\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + h \|D({{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}})\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \\ & \qquad +\max_{F \in {\mathcal F}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})} \sum_{r=0}^{k+1} h^{r} \|D^r ({{\it{\Theta}}_h^{\it{\Gamma}}} - {{\it{\Psi}}^{\it{\Gamma}}})\|_{\infty,F} +\max_{x_i \in {\mathcal V}(\partial {{\it{\Omega}}^{{\it{\Gamma}}}})}|({{\it{\Theta}}_h^{\it{\Gamma}}}-{{\it{\Psi}}^{\it{\Gamma}}})(x_i)| \\ &\quad \lesssim \sum_{r=0}^{k+1} h^r \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}}\Vert D^r({{\it{\Theta}}_h^{\it{\Gamma}}} - {\it{\Psi}}^{\it{\Gamma}}) \Vert_{\infty,T} \lesssim h^{k+1}. \end{align*} Note that $${\it{\Phi}}_h-{\rm id}=({\it{\Psi}}-{{\it{\Theta}}_h}){{\it{\Theta}}_h}^{-1}$$. From the results in (3.34) and (5.13), it follows that $$\|{{\it{\Theta}}_h}^{-1}\|_{\infty,{\it{\Omega}}} \lesssim 1$$, $$\|D{{\it{\Theta}}_h}^{-1}\|_{\infty,{\it{\Omega}}} \lesssim 1$$. Hence, the estimate in (5.14) follows from the one in (5.13). □ Remark 5.6 With similar arguments as used in the proof above, one can also derive the bound $\max_{T \in {\mathcal T}} \Vert D^l ({\it{\Phi}}_h - {\rm id}) \Vert_{\infty,T} = \max_{T \in {\mathcal T}} \Vert D^l {\it{\Phi}}_h \Vert_{\infty,T} \lesssim h^{k+1-l}, \qquad T \in {\mathcal T}, l=2,..,k+1,$ but we do not need this in the error analysis. 5.3 Strang lemma In this section, we derive a Strang lemma in which the discretization error is related to approximation and geometry errors. We use the homeomorphism $${\it{\Phi}}_h: {\it{\Omega}} \to {\it{\Omega}}$$ with the property $${\it{\Phi}}_h({\it{\Gamma}}_h)={\it{\Gamma}}$$. We define $${V_{\text{reg}}} :=H^1({\it{\Omega}})\cap H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$, cf. (5.4). Related to $${\it{\Phi}}_h$$ we define $V_{h,{\it{\Phi}}}^{\it{\Gamma}} := \{v \circ {\it{\Phi}}_h^{-1}, v \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}\} \subset H^1({\it{\Omega}}_1 \cup {\it{\Omega}}_2),$ and the linear and bilinear forms $$A(u,v) := a(u,v) + N(u,v), \quad f(v) := \sum_{i=1,2} \int_{{\it{\Omega}}_i} f_i v \, {\rm d}x, \quad u,v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}},$$ (5.15) with the bilinear forms $$a(\cdot,\cdot)$$, $$N(\cdot,\cdot)$$ as in (4.3) with $${\it{\Omega}}_{i,h}$$ and $${\it{\Gamma}}_h$$ replaced with $${\it{\Omega}}_i$$ and $${\it{\Gamma}}$$, respectively. In the following lemma, a consistency result is given. Lemma 5.7 Let $$u\in {V_{\text{reg}}}$$ be a solution of (1.1). The following holds: $$\label{eq:cons1} A(u,v) = f(v) \quad \text{for all} \,\, v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}}.$$ (5.16) Proof. Take $$v \in V_{h,{\it{\Phi}}}^{\it{\Gamma}} + {V_{\text{reg}}}$$. Since $$[\hspace{-2pt}[{u}]\hspace{-2pt}]_{|{\it{\Gamma}}}=0$$ a.e. on $${\it{\Gamma}}$$, we get $$N^s(u, v)= N^c(v,u)=0$$. Thus, we obtain \begin{align*} A(u,v)- f(v) & = a(u,v)+ N^c(u, v) - \sum_{i=1,2} \int_{{\it{\Omega}}_i} f_i v \, {\rm d}x \\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \alpha_i \nabla u \cdot \nabla v - f_i v \, {\rm d}x +\int_{{\it{\Gamma}}} \{\!\!\{{-\alpha \nabla u \cdot n}\}\!\!\} [\hspace{-2pt}[{v}]\hspace{-2pt}] \, {\rm d}s\\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \alpha_i \nabla u \cdot \nabla v - f v_i \, {\rm d}x+ \int_{{\it{\Gamma}}} [\hspace{-2pt}[{-\alpha \nabla u \cdot n v}]\hspace{-2pt}] \, {\rm d}s, \end{align*} where in the last equality we used the flux continuity in (1.1b). Applying partial integration on $${\it{\Omega}}_i$$ and using (1.1a) results in $A(u, v)- f(v)=\sum_{i=1}^2 \int_{{\it{\Omega}}_i}- {\textrm{div}\ \!} (\alpha_i \nabla u) v - f_i v \, {\rm d}x =0,$ which completes the proof. □ Lemma 5.8 Let $$u\in {V_{\text{reg}}}$$ be a solution of (1.1) and $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ the solution of (4.2). The following holds: $$\label{eq:strang} \begin{split} \Vert u \circ {\it{\Phi}}_h - u_h \Vert_h \lesssim & \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert u\circ {\it{\Phi}}_h - v_h \Vert_h \\ & + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| f_h(w_h) - f(w_h \circ {\it{\Phi}}_h^{-1}) |}{\Vert w_h \Vert_h} \\ & + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| A_h(u\circ {\it{\Phi}}_h,w_h) - A(u,w_h \circ {\it{\Phi}}_h^{-1}) |}{\Vert w_h \Vert_h}. \end{split}$$ (5.17) Proof. The proof is along the same lines as in the well-known Strang Lemma. We use the notation $$\tilde u =u \circ {\it{\Phi}}_h$$ and start with the triangle inequality, where we use an arbitrary $$v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: \begin{equation*} \Vert \tilde u - u_h \Vert_h \leq \Vert \tilde u - v_h \Vert_h + \Vert v_h - u_h \Vert_h. \end{equation*} With $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$-coercivity, cf. (5.8), we have $$\label{eq:strang2} \begin{split} \Vert u_h - v_h \Vert_h^2 & \lesssim A_h(u_h - v_h,u_h - v_h) \stackrel{w_h=u_h-v_h}{=} A_h(u_h-v_h,w_h) \\ & = \left| A_h(\tilde u-v_h,w_h) \right| + \left| A_h(\tilde u,w_h) - f_h(w_h) \right|\!. \end{split}$$ (5.18) Using continuity (5.9) and dividing by $$\Vert w_h \Vert_h = \Vert u_h - v_h \Vert_h$$ results in \begin{equation*} \Vert \tilde u - u_h \Vert_h \lesssim \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert \tilde u - v_h \Vert_h + \sup_{w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \frac{| A_h(\tilde u,w_h) - f_h(w_h) |}{\Vert w_h \Vert_h}. \end{equation*} Using the consistency property of Lemma 5.7 yields \begin{align*} & |A_h(\tilde u,w_h) - f_h(w_h)| \\ & \quad = |A_h(\tilde u,w_h) - f_h(w_h) - \underbrace{\left(A(u,w_h \circ {\it{\Phi}}_h^{-1}) - f(w_h\circ {\it{\Phi}}_h^{-1})\right)}_{=0}| \\ & \quad \leq |A_h(\tilde u, w_h) - A(u, w_h \circ {\it{\Phi}}_h^{-1})| + |f_h(w_h) - f(w_h \circ {\it{\Phi}}_h^{-1})| \end{align*} which completes the proof. □ Remark 5.9 In practice, we will commit an additional variational crime due to the numerical integration of integrands as in (4.5), which are in general not polynomial. We briefly discuss how to incorporate the related additional error terms in the error analysis without going into all the technical details. The procedure is essentially the same as in Sections 25–29 in Ciarlet (1991). We consider $$A_h^q$$ and $$f_h^q$$ the bilinear and linear forms which approximate $$A_h$$ and $$f_h$$ with numerical integration. The solution obtained by quadrature is denoted as $$u_h^q \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, which satisfies $$A_h^q(u_h^q,v_h) = f_h^q(v_h)$$ for all $$v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. For the numerical integration, we consider quadrature rules that have a certain exactness degree on the (cut) elements in the reference geometries $${\it{\Omega}}^{\text{lin}}_i$$ and $${\it{\Gamma}}^{\text{lin}}$$. For the integral in (4.5), this would be a quadrature of exactness degree $$2k-2$$. The additional terms $$D{{\it{\Theta}}_h}^{-T}$$ and $$\mathrm{det}(D{{\it{\Theta}}_h})$$ stemming from the transformation are close to $${\rm id}$$ and $$1$$, respectively, cf. the estimates in Lemma 3.14. As a consequence, we have that $$A_h(u,u) \sim A_h^q(u,u)$$ on $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ and thus we get coercivity of $$A_h^q(\cdot,\cdot)$$ w.r.t. $$\|\cdot\|_h$$ on $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$. The arguments in the proof of the Strang Lemma 5.8 apply with $$A_h(\cdot,\cdot)$$ replaced by $$A_h^q(\cdot,\cdot)$$. Hence, we have to bound consistency terms as in (5.17) with $$A_h(\cdot,\cdot)$$, $$f_h$$ replaced by $$A_h^q(\cdot,\cdot)$$ and $$f_h^q$$, respectively. This can be done by combining the techniques used in the proof of Lemma 5.10 below with the ones from Ciarlet (1991). 5.4 Consistency errors We derive bounds for the consistency error terms on the right-hand side in the Strang estimate (5.17). Lemma 5.10 Let $$u \in {V_{\text{reg}}}$$ be a solution of (1.1). We assume $$f \in H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ and the data extension source term $$f_h$$ in (4.4) satisfies $$\|f_h\|_{H^{1,\infty}({\it{\Omega}}_{1,h}\cup {\it{\Omega}}_{2,h})} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$. The following estimates hold for $$w_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$: \begin{align} |A(u,w_h\circ {\it{\Phi}}_h^{-1}) - A_h(u\circ {\it{\Phi}}_h,w_h)| & \lesssim h^k \Vert u \Vert_{H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \Vert w_h \Vert_h \label{p1} \\ \end{align} (5.19a) \begin{align} |f(w_h\circ {\it{\Phi}}_h^{-1})-f_h(w_h)| & \lesssim h^{k} \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} \| w_h \Vert_h \label{p2}. \end{align} (5.19b) Proof. The proof, which is elementary, is given in the Appendix A. □ 5.5 Approximation error In this section, we derive a bound for the approximation error, i.e., the first term on the right-hand side in the Strang estimate (5.17). For this, we use the curved finite element space $$V_{h,{\it{\Psi}}}=\{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^k\,\}$$ introduced in (3.36) and the corresponding optimal interpolation operator $${\it{\Pi}}_h$$, cf. (3.37). We also use the corresponding unfitted finite element space $$V_{h,{\it{\Psi}}}^{\it{\Gamma}}:=\{\, v_h \circ {\it{\Psi}}_h^{-1} | v_h \in V_h^{\it{\Gamma}}\,\}$$, cf. (4.1). Lemma 5.11 For $$u \in H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ the following holds: $$\label{approxerrro} \inf_{v_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}} \Vert u\circ {\it{\Phi}}_h - v_h \Vert_h =\inf_{v_h \in V_{h,{\it{\Psi}}}^{\it{\Gamma}}} \Vert (u- v_h)\circ {\it{\Phi}}_h \Vert_h \lesssim h^k \|u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}.$$ (5.20) Proof. The identity in (5.20) follows from the definitions of the spaces $$V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$, $$V_{h,{\it{\Psi}}}^{\it{\Gamma}}$$. The analysis is along the same lines as known in the literature (e.g., Hansbo & Hansbo, 2002; Reusken, 2008). Let $${\it{\Pi}}_h$$ be the interpolation operator in $$V_{h,{\it{\Psi}}}$$ and $$R_i$$ the restriction operator $$R_iv:=v_{|{\it{\Omega}}_{i}}$$. We use the bounded linear extension operators $$E_i:\, H^{k+1}({\it{\Omega}}_i) \to H^{k+1}({\it{\Omega}})$$, cf., for example, Theorem II.3.3 in Galdi (2011) and define $$u_i^e := E_i u_i, i=1,2$$. We define $$v_h := \sum_{i=1}^2 R_i {\it{\Pi}}_h u_i^e \in V_{h,{\it{\Psi}}}^{\it{\Gamma}}$$. Note that due to (3.37) we have a optimal interpolation error bounds for $$u_i^e \in H^{k+1}({\it{\Omega}})$$. We get: \begin{align*} |(u - v_h)\circ {\it{\Phi}}_h|_1^2& \lesssim \sum_{i=1}^2 \alpha_i \|\nabla (u_i - R_i {\it{\Pi}}_h u_i^e)\|_{L^2({\it{\Omega}}_{i})}^2 \\ & \lesssim \sum_{i=1}^2 \|\nabla (u_i^e - {\it{\Pi}}_h u_i^e)\|_{L^2({\it{\Omega}})}^2 \lesssim h^{2k} \sum_{i=1}^2 \| u_i^e \|_{H^{k+1}({\it{\Omega}})}^2. \end{align*} This yields the desired bound for the $$|\cdot|_1$$ part of the norm $$\|\cdot\|_h$$. As in Hansbo & Hansbo (2002), we obtain \begin{align*} \Vert [\hspace{-2pt}[{(u- v_h)\circ {\it{\Phi}}_h}]\hspace{-2pt}] \Vert_{\frac12,h,{\it{\Gamma}}_h} &= \left(\bar \alpha / h \right)^\frac12 \|[\hspace{-2pt}[{(u - v_h)\circ {\it{\Phi}}_h}]\hspace{-2pt}]\|_{L^2({\it{\Gamma}}_h)} \lesssim h^{- \frac12} \sum_{i=1}^2 \| u_i^e - {\it{\Pi}}_h u_i^e \|_{L^2({\it{\Gamma}})} \\ & \lesssim \sum_{i=1}^2\Big(h^{-1} \| u_i^e - {\it{\Pi}}_h u_i^e \|_{L^2({\it{\Omega}})} +\| u_i^e - {\it{\Pi}}_h u_i^e \|_{H^1({\it{\Omega}})}\Big) \\ & \lesssim h^k \sum_{i=1}^2 \| u_i^e \|_{H^{k+1}({\it{\Omega}})} \lesssim h^k \sum_{i=1}^2 \| u_i \|_{H^{k+1}({\it{\Omega}}_i)}. \end{align*} Using very similar arguments, cf. (Hansbo & Hansbo, 2002), the same bound can be derived for the term $$\Vert \{\!\!\{{(u- v_h)\circ {\it{\Phi}}_h}\}\!\!\}\Vert_{-\frac12,h,{\it{\Gamma}}_h}$$. Thus, we get $$\|(u-v_h)\circ {\it{\Phi}}_h\|_h \lesssim h^k \| u\|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$, which completes the proof. □ 5.6 Discretization error bounds Based on the Strang estimate and the results derived in the subsections above, we immediately obtain the following theorem, which is the main result of this article. Theorem 5.12 Let $$u$$ be the solution of (1.1) and $$u_h \in V_{h,{\it{\Theta}}}^{\it{\Gamma}}$$ the solution of (4.2). We assume that $$u\in H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)$$ and the data extension $$f_h$$ satisfies the condition in Lemma 5.10. Then, the following holds: $$\label{main1} \Vert u\circ {\it{\Phi}}_h - u_h \Vert_h \lesssim h^k(\| u \|_{H^{k+1}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)} + \|f\|_{H^{1,\infty}({\it{\Omega}}_1\cup{\it{\Omega}}_2)}).$$ (5.21) Proof. The result directly follows from the Strang Lemma 5.8 and the bounds derived in Lemmas 5.10 and 5.11. □ 6. Discussion and outlook We presented a rigorous error analysis of a high-order unfitted FEM applied to a model interface problem. The key component in the method is a parametric mapping $${{\it{\Theta}}_h}$$, which transforms the piecewise linear planar interface approximation $${{\it{\Gamma}}^{\mathrm{lin}}}$$ to a higher-order approximation $${\it{\Gamma}}_h$$ of the exact interface $${\it{\Gamma}}$$. This mapping is based on a new local level set-based mesh transformation combined with an extension technique known from isoparametric finite elements. The corresponding isoparametric unfitted finite element space is used for discretization, and combined with a standard Nitsche technique for enforcing continuity of the solution across the interface in a weak sense. The discretization error analysis is based on a Strang Lemma. For handling the geometric error, a suitable bijective mapping $${\it{\Phi}}_h$$ on $${\it{\Omega}}$$ is constructed, which maps the numerical interface $${\it{\Gamma}}_h$$ to the exact interface $${\it{\Gamma}}$$, and is sufficiently close to the identity. The construction of this mapping is based on the same approach as used in the construction of $${{\it{\Theta}}_h}$$. The main results of the article are the optimal high-order ($$H^1$$-norm) error bounds given in Section 5.6. The derivation of an optimal order $$L^2$$-norm error bound, based on a duality argument, will be presented in a forthcoming article. For this, we need an improvement of the consistency error bound in (5.19a), which is of the order $$\mathcal{O}(h^k)$$, to a bound of order $$\mathcal{O}(h^{k+1})$$. The methodology presented in this article, especially the parametric mapping and the error analysis of the geometric errors, can also be applied in other settings with unfitted FEMs, for example, fictitious domain methods (Lehrenfeld, 2016b), unfitted FEM for Stokes interface problems (Lederer et al., 2016) and trace FEMs for surface PDEs (Grande et al., 2016). Appendix A A.1 Proof of Lemma 3.5 First, we prove the existence of a unique solution $$d_h(x)$$ of (3.12). For $$|\alpha| \leq \alpha_0 h$$, with $$\alpha_0 >0$$ fixed, we introduce the polynomial $p(\alpha):= \mathcal{E}_T \phi_h(x + \alpha G_h(x)) - \hat \phi_h (x)$ for a fixed $$x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. From (2.4) it follows that $$\label{estt} | \hat \phi_h (x) - \phi(x)| \lesssim h^2.$$ (A.1) Furthermore, with $$y:=x +\alpha G_h(x)$$, we have $$\|x-y\|_2 \lesssim h$$, and using a Taylor expansion and (2.3), we obtain (for $$\xi \in (0,1)$$) $$\label{Taylor} \begin{split} \big| (\mathcal{E}_T \phi_{h})(y)-\phi(y)\big| = & \ \Bigg| \sum_{m=0}^k \frac{1}{m!} D^m(\mathcal{E}_T \phi_{h}-\phi)(x)(y-x, \ldots,y-x) \\ & \ + \frac{1}{(k+1)!}D^{k+1}\phi(x+\xi(y-x))(y-x,\ldots,y-x)\Bigg| \\ \lesssim & \sum_{m=0}^k |\phi_h -\phi|_{m, \infty,T} \|y-x\|_2^m + \|y-x\|_2^{k+1} \lesssim h^{k+1}. \end{split}$$ (A.2) Thus, we get $$p(\alpha)= \phi(x+ \alpha G_h(x))- \phi(x) +\mathcal{O}(h^2)$$, where the constant in $$\mathcal{O}(\cdot)$$ is independent of $$x$$. Using Lemma 3.3 and (2.2), we obtain $$\label{lk} p(\alpha)= \alpha \|\nabla \phi(x)\|_2^2 + \mathcal{O}(h^2).$$ (A.3) Hence, there exists $$h_0 >0$$, such that for all $$0< h\leq h_0$$ and all $$x\in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, the equation $$p(\alpha)=0$$ has a unique solution $$\alpha =:d_h(x)$$ in $$[-\alpha_0 h,\alpha_0 h]$$. The result in (3.13b) follows from (A.3). The smoothness property, $$d_h \in C^\infty(T)$$, $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$, follows from the fact that $$G_h$$ and $$\hat \phi_h$$ are polynomials on $$T$$, and $$\mathcal{E}_T \phi_h$$ is a global polynomial. We note that for each vertex $$x_i$$ of $$T \in {\mathcal T}$$, we have $$\mathcal{E}_T \phi_h(x_i) = \phi_h(x_i) = {\hat \phi_h}(x_i)$$ and it follows that $$d_h(x_i)=0$$ solves (3.12), and hence (3.13a) holds. We finally consider (3.13c). We differentiate the relation (3.12) on $$T$$. We skip the argument $$x$$ in the notation and use $$y_h:=x+d_h(x) G_h(x)$$: $$\label{kjj} \nabla(\mathcal{E}_T\phi_h)(y_h) + \nabla d_h \nabla (\mathcal{E}_T\phi_h)(y_h)^TG_h + d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h) = \nabla \hat \phi_h.$$ (A.4) Using a Taylor expansion as in (A.2) we get $$|\nabla(\mathcal{E}_T\phi_h)(y_h)-\nabla \phi(y_h)| \lesssim h^k$$. Using (3.13b), (3.11) and rearranging terms in (A.4), we get (for $$k \geq 2$$): $\nabla d_h \big(\|\nabla \phi\|_2^2 + \mathcal{O}(h^2)\big)= \nabla \hat \phi_h - \nabla(\mathcal{E}_T\phi_h)(y_h)-d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h).$ From the estimates derived above and the smoothness assumption on $$\phi$$, it easily follows that the second and third term on the right-hand side are uniformly bounded by a constant and by $$ch^2$$, respectively. We further have $$\nabla {\hat \phi_h} = \nabla \phi + \nabla {\hat \phi_h} - \nabla \phi$$ with $$\Vert \nabla {\hat \phi_h} - \nabla \phi \Vert_{\infty} \lesssim h$$ and $$\Vert \nabla \phi \Vert_{\infty} \lesssim 1$$. Combining these results completes the proof. A.2 Proof of Lemma 3.6 Recall that $${\it{\Psi}}^{\it{\Gamma}}(x)\,{=}\, x+ d(x) G(x)$$, $${\it{\Psi}}_h^{\it{\Gamma}}(x)\,{=}\,x+d_h(x)G_h(x)$$. We have $$\|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \,{\lesssim}\, h^2, \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d_h\|_{\infty,T} \,{\lesssim}\, h^2$$, cf. (3.2b) and (3.13b). Take $$x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. We start with the triangle inequality $$\label{ttt} \begin{split} \|({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}})(x)\|_2 & = \| d(x) G(x)- d_h(x)G_h(x)\|_2 \\ & \leq |d(x)-d_h(x)| \|G\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} + |d_h(x)| \, \|G_h(x)-G(x)\|_2. \end{split}$$ (A.5) The second term on the right-hand side is uniformly bounded by $$h^{k+2}$$ due to the result in Lemma 3.3. We have by construction $$\label{phiphih} \phi(x+ d(x) G(x))= \hat\phi_h(x)=(\mathcal{E}_T\phi_{h}) (x+ d_h(x) G_h(x)), x \in T \in {{\mathcal T}^{{\it{\Gamma}}}}.$$ (A.6) Using this and (2.2) we get \begin{align*} |d(x)-d_h(x)| & \sim \big| \phi(x+ d(x) G(x)) -\phi(x+ d_h(x) G(x))\big| \\ & =\big| (\mathcal{E}_T \phi_{h})(x+ d_h(x) G_h(x))-\phi(x+ d_h(x) G(x))\big| \\ & \leq \big| (\mathcal{E}_T \phi_{h}) (x+ d_h(x) G_h(x))-\phi(x+ d_h(x) G_h(x))\big| \\ & \quad +\big|\phi(x+ d_h(x) G_h(x))- \phi(x+ d_h(x) G(x))\big|. \end{align*} Using the estimate in (A.2), it follows that the first term on the right-hand side is uniformly bounded by $$h^{k+1}$$. The second term on the right-hand side is uniformly bounded by $\big|\phi(x+ d_h(x) G_h(x))- \phi(x+ d_h(x) G(x))\big| \lesssim |d_h(x)| \, \|G_h(x)- G(x)\|_2 \lesssim h^{k+2}.$ Collecting these results we obtain $$\label{NN} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|d-d_h\|_{\infty,T} \lesssim h^{k+1},$$ (A.7) which yields the desired bound for the first term on the right-hand side in (A.5). Hence, the bound for the first term in (3.16) is proven. It remains to bound the derivatives. We take the derivative in the equation (A.6). To simpifly the notation, we drop the argument $$x$$ and set $$y:=x+d(x)G(x)$$, $$y_h:=x+d_h(x)G_h(x)$$. This yields $$\label{kj} \begin{split} & \nabla \phi(y) + \nabla d \nabla \phi(y)^T \nabla \phi + d DG^T \nabla \phi(y) \\ &\quad = \nabla(\mathcal{E}_T\phi_h)(y_h) + \nabla d_h \nabla (\mathcal{E}_T\phi_h)(y_h)^T G_h + d_h DG_h^T \nabla (\mathcal{E}_T\phi_h)(y_h). \end{split}$$ (A.8) Using a Taylor expansion as in (A.2) we get $$\label{Tay} \|\nabla(\mathcal{E}_T\phi_h)(y_h)-\nabla \phi(y_h)\|_2 \lesssim h^k.$$ (A.9) Furthermore, $$\|\nabla \phi(y)- \nabla \phi(x)\|_2 \lesssim h^2$$, $$\|\nabla \phi(y_h)- \nabla \phi(x)\|_2 \lesssim h^2$$ and $$\|G_h(x)-\nabla \phi(x)\|_2 \lesssim h$$. Using these results and (3.11) and rearranging terms in (A.8), we get \begin{align*} & \big(\nabla d - \nabla d_h\big)\big(\|\nabla \phi\|_2^2 + \mathcal{O}(h)\big) \\ &\quad = \big(\nabla (\mathcal{E}_T\phi_h)(y_h)- \nabla \phi(y)\big) +\big(d_h DG_h^T \nabla(\mathcal{E}_T\phi_h)(y_h) - d DG^T \nabla \phi(y) \big). \end{align*} Using (3.11) and (A.7), we get $$|y-y_h| \lesssim h^{k+1}$$. Using this and (A.9), we obtain for the first term on the right-hand side the uniform bound $$c h^k$$. For the second term, we obtain the same uniform bound using (A.9), (A.7), (3.11) and $$\max_{T \in {\mathcal T}}\|d_h\|_{\infty,T} \lesssim h^2$$, $$\|d\|_{\infty,{{\it{\Omega}}^{{\it{\Gamma}}}}} \lesssim h^2$$. This yields $$\label{re1} \max_{T \in {{\mathcal T}^{{\it{\Gamma}}}}} \|\nabla d - \nabla d_h\|_{\infty,T} \lesssim h^k.$$ (A.10) Using this, (A.7) and the results in Lemmas 3.3 and 3.5, we obtain the estimates \begin{align*} & \Vert D ({\it{\Psi}}^{\it{\Gamma}} - {\it{\Psi}}_h^{\it{\Gamma}}) \Vert_{\infty,T} = \Vert D (d G - d_h G_h) \Vert_{\infty,T} \\ &\quad \lesssim \Vert \nabla (d-d_h) \Vert_{\infty, T} \Vert G \Vert_{\infty,T} + \Vert \nabla d_h \Vert_{\infty, T} \Vert G - G_h \Vert_{\infty, T} \\ & \qquad + \|d-d_h\|_{\infty, T} \Vert D G \Vert_{\infty,T} + \|d_h\|_{\infty, T} \Vert D(G-G_h) \Vert_{\infty, T} \lesssim h^k, \end{align*} which are uniform in $$T \in {{\mathcal T}^{{\it{\Gamma}}}}$$. A.3 Proof of Lemma 3.10 Recall that $$\mathcal{E}^{F\rightarrow T} w := \mathcal{E}^{\hat F\rightarrow \hat T} (w \circ \hat {\it{\Phi}}_F^{-1}) \circ {\it{\Phi}}_T^{-1}$$, $$\hat{w} = w \circ \hat {\it{\Phi}}_F^{-1}$$. We transform from $$T$$,$$F$$ to the corresponding reference element and face, and use $$\Vert D^n \mathcal{E}^{F \rightarrow T} w \Vert_{\infty, T} \lesssim h^{-n} \Vert D^n \mathcal{E}^{\hat F \rightarrow \hat T} \hat w \Vert_{\infty, \hat{T}}, \quad \Vert D^r \hat{w} \Vert_{\infty, \hat{F}} \lesssim h^r \Vert D^r w \Vert_{\infty, F}.$$ From this it follows that it suffices to prove $$\label{transfeq} \Vert D^n \mathcal{E}^{\hat F\rightarrow \hat T} \hat{w}_T \Vert_{\infty,\hat{T}} \lesssim \sum_{r=n}^{k+1} \Vert D^r \hat{w} \Vert_{\infty, \hat{F}}, \quad \forall \hat w \in C_0^{k+1}(\hat F), n=0,\ldots,k+1.$$ (A.11) Recall that $$\label{defE} \mathcal{E}^{\hat{F}\rightarrow \hat{T}} \hat w :=\omega^{k+1} A_k^\ast(\hat w)\circ Z+ \sum_{l=2}^k \omega^l A_l(\hat{w}) \circ Z,$$ (A.12) with $$A_l^\ast := {\rm id} - {\it{\Lambda}}_l, \quad A_l := {\it{\Lambda}}_l - {\it{\Lambda}}_{l-1} = - A_{l}^\ast + A_{l-1}^\ast$$, $${\it{\Lambda}}_l: C(\hat{F}) \rightarrow \mathcal{P}^l(\hat{F})$$ the interpolation operator, and $$\omega$$, $$Z$$ as in (3.18). The proof uses techniques also used in Bernardi (1989). We apply a standard Bramble–Hilbert argument, which yields $$\label{alstar} \Vert D^r (A_l^\ast \hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}}, \quad 0\leq r \leq l+1 \leq k+1,$$ (A.13) and thus, by a triangle inequality, $$\label{lstar} \Vert D^r (A_l \hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}} + \Vert D^{l} \hat{w} \Vert_{\infty,\hat{F}}, \quad 0\leq r \leq l+1 \leq k+1, l \geq 2.$$ (A.14) For $$Z$$, we have $$\Vert D^m Z \Vert_{\infty,\hat{T}} \lesssim \omega^{-m}, m \geq 1$$, cf. (Bernardi, 1989, Lemma 6.1). With a higher-order multivariate chain rule, cf. (Ciarlet & Raviart, 1972), we further have \begin{align} \Vert D^m A_l^\ast(\hat{w}) \circ Z \Vert_{\infty,\hat{T}} & \lesssim \sum_{r=1}^m \underbrace{\Vert D^r A_l^\ast(\hat{w}) \Vert_{\infty,\hat{F}}}_{\lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty, \hat{F}}} \sum_{\alpha \in \mathcal{J}(r,m)} \underbrace{\prod_{q=1}^m \Vert D^q Z \Vert^{\alpha_q}}_{\omega^{-m}} \\ \end{align} (A.15) \begin{align} & \lesssim \Vert D^{l+1}\hat w \Vert_{\infty,\hat{F}} \, \omega^{-m}, \quad 1 \leq m \leq l+1\leq k+1, \label{kll} \\ \text{where} \quad \mathcal{J}(r,m) & := \left\{\alpha = (\alpha_1,..,\alpha_r) \mid \sum_{q=1}^m \alpha_q = r, \sum_{q=1}^m q \alpha_q = m\right\}\!, \nonumber \end{align} (A.16) and for $$m=0$$ $$\Vert A_l^\ast(\hat{w}) \circ Z \Vert_{\infty,\hat{T}} = \Vert A_l^\ast(\hat{w}) \Vert_{\infty,\hat{F}} \lesssim \Vert D^{l+1} \hat{w} \Vert_{\infty,\hat{F}}.$$ Similarly, using $$D^mA_l(\hat w)=0$$ for $$m >l$$, $\Vert D^m A_l (\hat{w}) \circ Z \Vert_{\infty,\hat{T}} \lesssim \left(\Vert D^{l+1} \hat{w} \Vert_{\infty, \hat{F}} + \Vert D^{l} \hat{w} \Vert_{\infty, \hat{F}} \right) \omega^{-m}, 0 \leq m, l\leq k+1, l \geq 2.$ Using a Bramble–Hilbert argument, Leibniz formula and the bound derived in (A.16) we get, for $$n \leq k+1$$, \begin{align*} \|D^n(\omega^{k+1} A_k^\ast(\hat w)\circ Z)\|_{\infty,\hat F} & \lesssim \sum_{r=0}^{n} \omega^{k+1-r} \underbrace{|D \omega|^r}_{\lesssim 1} \Vert D^{n-r} (A_k^\ast(\hat w \circ Z)) \Vert_{\infty,\hat T} \\ & \lesssim \sum_{r=0}^{k+1} \omega^{k+1-r} \|D^{k+1}\hat w\|_{\infty,\hat F}\,\omega^{-(n-r)} \lesssim \|D^{k+1}\hat w\|_{\infty,\hat F}, \end{align*} which yields the desired bound for the first term on the right-hand side in (A.12). We now consider one term $$\omega^l A_l(\hat w)\circ Z$$, $$2 \leq l \leq k$$, from the second term on the right-hand side in (A.12). This is a polynomial of degree $$l$$, hence $$D^n(\omega^l A_l(\hat w)\circ Z) =0$$ for $$n >l$$. We have to consider only $$n \leq l \leq k$$. With similar arguments as above we get: \begin{align} \Vert D^n (\omega^l A_l(\hat w)\circ Z) \Vert_{\infty,\hat T} & \lesssim \sum_{r=0}^n \omega^{l-r} |D \omega|^r \Vert D^{n-r} (A_l(\hat w \circ Z)) \Vert_{\infty,\hat T} \\ \end{align} (A.17) \begin{align} &\lesssim \sum_{r=0}^{k+1} \underbrace{\omega^{l-r} \omega^{-(n-r)}}_{\lesssim 1 (l \geq n)} (\Vert D^l \hat w \Vert_{\infty,\hat F} + \Vert D^{l+1} \hat w \Vert_{\infty,\hat F}). \end{align} (A.18) Summing over $$l=2, \ldots, k$$ completes the proof. A.4 Proof of Lemma 5.10 We first consider the terms $$a_h(\cdot,\cdot)$$, $$a(\cdot,\cdot)$$ in $$A_h(\cdot,\cdot)$$ and $$A(\cdot,\cdot)$$, respectively, split the bilinear forms into its contributions from the two subdomains, cf. (4.5), for instance $$a_h(\cdot,\cdot) = a_h^1(\cdot,\cdot) + a_h^2(\cdot,\cdot)$$, and consider one part $$a_h^i(\cdot,\cdot)$$, $$i=1,2$$. We introduce $$\tilde w_h:= w_h \circ {\it{\Phi}}_h^{-1}$$ and $$\tilde u:= u \circ {\it{\Phi}}_h$$, and compare the reference formulation without geometrical errors \begin{align*} a^i(u, \tilde w_h) &= \alpha_i \int_{{\it{\Omega}}_i} \nabla u \cdot \nabla \tilde w_h \, {\rm d}x = \alpha_i \int_{{\it{\Omega}}_{i,h}} \det(D {\it{\Phi}}_h) D {\it{\Phi}}_h^{-T} \nabla \tilde u \cdot D {\it{\Phi}}_h^{-T} \nabla w_h \, {\rm d}y \end{align*} with the discrete variant \begin{align*} a_h^i(\tilde u,w_h) & = \alpha_i \int_{{\it{\Omega}}_{i,h}} \nabla \tilde u \cdot \nabla w_h \, {\rm d}y. \end{align*} From this we get $|a^i(u, \tilde w_h) - a_h^i(\tilde u,w_h)| = \alpha_i \left| \int_{{\it{\Omega}}_{i,h}} \nabla \tilde u^T \cdot C \nabla w_h \, {\rm d}y\right| \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\tilde u\|_{H^1({\it{\Omega}}_{i,h})}\|w_h\|_h,$ with $$C:= \det(A) (A^TA)^{-1}-I$$, $$A:=D{\it{\Phi}}_h$$. From Lemma 5.5, we have $$\|A-I\|_{\infty,{\it{\Omega}}} \,{\lesssim}\, h^k \text{and} \|A^{-1}-I\|_{\infty,{\it{\Omega}}} \,{\lesssim}\, h^k$$, which implies $$\Vert \mathrm{det}(A) - 1 \Vert_{\infty,{\it{\Omega}}} \lesssim h^k$$. We thus obtain the bound $$\Vert C \Vert_{\infty,{\it{\Omega}}} \lesssim \Vert \mathrm{det}(A) - 1 \Vert_{\infty,{\it{\Omega}}} + \Vert A - I \Vert_{\infty,{\it{\Omega}}} + \Vert A^{-1} - I \Vert_{\infty,{\it{\Omega}}} \lesssim h^k.$$ Combining these estimates results in $$\label{est1} |a^i(u, \tilde w_h) - a_h^i(\tilde u,w_h)| \lesssim h^k \Vert \tilde u \Vert_{H^1({\it{\Omega}}_{i,h})} \Vert w_h \Vert_h.$$ (A.19) With similar arguments we obtain the estimate $$\Vert \tilde u \Vert_{H^{1}({\it{\Omega}}_{i,h})} \sim \Vert u \Vert_{H^{1}({\it{\Omega}}_i)}.$$ We now consider the terms $$N_h^c(\tilde u, w_h)$$ and $$N^c(\tilde u, w_h)$$. We use a relation between $$n_{{\it{\Gamma}}_h}(x)=:n_{{\it{\Gamma}}_h}$$ and $$n_{\it{\Gamma}}({\it{\Phi}}_h(x))$$, $$x \in {\it{\Gamma}}_h \cap {{\it{\Theta}}_h}(T)={{\it{\Theta}}_h}({{\it{\Gamma}}^{\mathrm{lin}}_T})$$. Let $$t_1, \ldots, t_{d-1}$$ be an orthonormal basis of $$n_{{\it{\Gamma}}_h}^\perp$$. The tangent space to $${\it{\Gamma}}= {\it{\Phi}}_h({\it{\Gamma}}_h)$$ at $${\it{\Phi}}_h(x)$$ is given by $${\rm span}\{\, D{\it{\Phi}}_h(x)t_j | 1 \leq j \leq d-1\,\}=n_{\it{\Gamma}}({\it{\Phi}}_h(x))^\perp$$. Hence, $$n_{\it{\Gamma}}({\it{\Phi}}_h(x)) \in {\rm span}\{\, D{\it{\Phi}}_h(x)^{-T}n_{{\it{\Gamma}}_h}\,\}$$ holds. This yields $$\label{traans} n_{\it{\Gamma}} \circ {\it{\Phi}}_h = \frac{D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h}}{\Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2}.$$ (A.20) In the transformation from $${\it{\Gamma}}_h$$ to $${\it{\Gamma}}$$, we get the factor $$\det(D{\it{\Phi}}_h) \Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2$$ from the change in the measure, cf. (3.40). The term $$\Vert D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \Vert_2$$ is the same as the denominator in the change of normal directions (A.20). Therefore, these terms cancel and simplify the formulas below. With the same matrix $$C$$ as used above, we get \begin{align} & \big| N^c(u, \tilde w_h) - N_h^c(\tilde u, w_h) \big| \nonumber \\ &\quad = \left|\int_{{\it{\Gamma}}_h} \det (D {\it{\Phi}}_h) \, D{\it{\Phi}}_h^{-T} \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} \cdot D{\it{\Phi}}_h^{-T} n_{{\it{\Gamma}}_h} \ [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s - \int_{{\it{\Gamma}}_h} \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} \cdot n_{{\it{\Gamma}}_h} \ [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s\right| \nonumber \\ &\quad = \left| \int_{{\it{\Gamma}}_h} n_{{\it{\Gamma}}_h}^T \cdot C \{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\} [\hspace{-2pt}[{w_h}]\hspace{-2pt}] \, {\rm d}s\right| \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\{\!\!\{{-\alpha \nabla \tilde u}\}\!\!\}\|_{-\frac12,h,{\it{\Gamma}}_h} \|[\hspace{-2pt}[{w_h}]\hspace{-2pt}]\|_{\frac12,h,{\it{\Gamma}}_h} \nonumber \\ &\quad \lesssim \|C\|_{\infty,{\it{\Omega}}} \|\{\!\!\{{-\alpha \nabla u}\}\!\!\}\|_{-\frac12,h,{\it{\Gamma}}} \|w_h\|_h \lesssim h^{k+\frac12} \|u\|_{H^2({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}\|w_h\|_h. \label{pl} \end{align} (A.21) Finally, we use $$[\hspace{-2pt}[{\tilde u}]\hspace{-2pt}] = 0$$ at $${\it{\Gamma}}_h$$ to note that $$N_h^c(w_h, \tilde{u})=N_h^s(\tilde{u},w_h)=N^c(\tilde w_h, u)=N^s(u, \tilde w_h)=0$$. Combining this with the bounds in (A.19), (A.21) and the definition of $$A_h(\cdot,\cdot)$$ and $$A(\cdot,\cdot)$$, we obtain the bound in (5.19a). For the other consistency term in the Strang estimate we obtain: \begin{align*} |f(\tilde w_h)-f_h(w_h)| & \leq \sum_{i=1,2} \left| \int_{{\it{\Omega}}_{i}} f_i (w_h\circ {\it{\Phi}}_h^{-1}) \, {\rm d}x - \int_{{\it{\Omega}}_{i,h}} f_{i,h} w_h\, {\rm d}x \right|\!, \end{align*} and for $$i=1,2$$ we get, using $$f_{i,h}=f_i$$ on $${\it{\Omega}}_i$$, \begin{align*} & \left| \int_{{\it{\Omega}}_{i}} f_i (w_h\circ {\it{\Phi}}_h^{-1}) \, {\rm d}x - \int_{{\it{\Omega}}_{i,h}} f_{i,h} w_h\, {\rm d}x \right| \\ &\quad \leq \left|\int_{{\it{\Omega}}_{i,h}} (\det(D{\it{\Phi}}_h) (f_{i,h} \circ {\it{\Phi}}_h) - f_{i,h}) w_h \, {\rm d}x \right|\\ &\quad \lesssim \|f_{i,h} \circ {\it{\Phi}}_h - f_{i,h} \|_{\infty,{\it{\Omega}}_{i,h}} \|w_h\|_h + {\Vert} \det(D{\it{\Phi}}_h) -1 \Vert_{\infty,{\it{\Omega}}_{i,h}} \|f_i\|_{\infty,{\it{\Omega}}_i} \|w_h\|_h. \end{align*} Finally note that $${\Vert}\det(D{\it{\Phi}}_h) -1 \Vert_{\infty,{\it{\Omega}}} \lesssim h^k$$ and $$\|f_h\|_{H^{1,\infty}({\it{\Omega}}_{1,h}\cup {\it{\Omega}}_{2,h})} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1 \cup {\it{\Omega}}_2)}$$, and thus we get: $\|f_{i,h} \circ {\it{\Phi}}_h - f_{i,h}\|_{\infty,{\it{\Omega}}_{i,h}} \leq \|f_{i,h}\|_{H^{1,\infty}({\it{\Omega}}_{i,h})} \|{\it{\Phi}}_h - {\rm id}\|_{\infty, {\it{\Omega}}} \lesssim \|f\|_{H^{1,\infty}({\it{\Omega}}_1\cup{\it{\Omega}}_2)} h^{k+1}$ which completes the proof of (5.19b). 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Published: Aug 18, 2017

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