On the DPG method for Signorini problems

On the DPG method for Signorini problems Abstract We derive and analyse discontinuous Petrov–Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and nonsymmetric formulations, where optimal test functions are used only for the partial differential equation part of the problem, not the boundary conditions. For the symmetric case and lowest-order approximations, we provide a simple a posteriori error estimate. In the second part, we apply our technique to the singularly perturbed case of reaction-dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case. 1. Introduction In this article, we develop a framework to solve contact problems by the discontinuous Petrov–Galerkin method with optimal test functions (DPG method). We consider a simplified model problem ($$-{\it{\Delta}} u+u=f$$ in a bounded domain) with unilateral boundary conditions that resembles a scalar version of the Signorini problem in linear elasticity (Signorini, 1933; Fichera, 1971). We prove well-posedness of our formulation and quasi-optimal convergence. We then illustrate how the scheme can be adapted to the singularly perturbed case of reaction-dominated diffusion ($$-\varepsilon{\it{\Delta}} u + u=f$$). Specifically, we use the DPG setting from Heuer & Karkulik (2017) to design a method that controls the field variables ($$u$$, $$\nabla u$$ and $${\it{\Delta}} u$$) in the so-called balanced norm (which corresponds to $$\|u\|+\varepsilon^{1/4}\|\nabla u\|+\varepsilon^{3/4}\|{\it{\Delta}} u\|$$ with $$L^2$$-norm $$\|\cdot\|$$; Lin & Stynes, 2012). The balanced norm is stronger than the energy norm that stems from the Dirichlet bilinear form of the problem. There is a long history of numerical analysis for contact problems and, more generally, for variational inequalities; cf. the books by Glowinski et al. (1981) and Kikuchi & Oden (1988). Some early articles on the (standard and mixed) finite element method for Signorini-type problems are Hlaváček & Lovíšsek (1977), Brezzi et al. (1977), Scarpini & Vivaldi (1977), Brezzi et al. (1978) and Loppin (1978); see also Gwinner (2013) for a more recent work. Unilateral boundary conditions usually give rise to limited regularity of the solution, and authors have made an effort to establish optimal convergence rates of the finite element method (see, e.g., Chouly & Hild, 2013; Drouet & Hild, 2015). Recently, there has been some interest in extending other numerical schemes to unilateral contact problems, e.g., least squares (Attia et al., 2009), the local discontinuous Galerkin method (Bustinza & Sayas, 2012) and a Nitsche-based finite element method (Chouly & Hild, 2013). Our objective is to set up an appropriate framework to apply DPG technology to variational inequalities, in this case Signorini-type problems. At this point, we do not intend to be more competitive than previous schemes. In particular, we are not concerned with the reduced regularity of solutions that limit convergence orders. In any case, we show well-posedness of our scheme for the minimum regularity of $$u$$ in $$H^1$$ and right-hand-side function $$f$$ in $$L^2$$. Our primary focus is to use DPG schemes in such a way that their performance is not reduced by the presence of Signorini boundary conditions. The DPG method aims at ensuring discrete inf–sup stability by the choice of norms and test functions (cf. Demkowicz & Gopalakrishnan, 2011a,b). This is particularly important for singularly perturbed problems where one can achieve robustness of error control (the discrete inf–sup constant does not depend on perturbation parameters; see Demkowicz & Heuer, 2013; Niemi et al., 2013; Broersen & Stevenson, 2014; Chan et al., 2014; Heuer & Karkulik, 2017). Recently, we have found a setting for the coupling of the DPG method and the Galerkin boundary element method (BEM) to solve transmission problems (see Führer et al., 2017a). The principal idea is to take a variational formulation of the interior problem (suitable for the DPG method) and to add transmission conditions as a constraint. This constraint can be given as boundary integral equations in a least-squares or Galerkin form. In this article, we follow this very strategy. The partial differential equation (PDE) is put in variational form (without considering any boundary condition) and the Signorini conditions are added as a constraint. It turns out that the whole scheme can be written as a variational inequality of the first kind where only the PDE part is tested with optimal test functions, as is the DPG strategy. Then, proving coercivity and boundedness of the bilinear form, the Lions–Stampacchia theorem proves well-posedness. Let us note that there are finite element/boundary element coupling schemes for contact problems (see Carstensen & Gwinner, 1997; Maischak & Stephan, 2005; Gatica et al., 2011). However, their coupling variants generalize the setting of contact problems on bounded domains. In contrast, our coupling scheme (Führer et al., 2017a) (for a standard transmission problem) separates the PDE on the bounded domain from the transmission conditions (and exterior problem) in such a way that they can be formulated as a variational PDE plus constraint. As we have mentioned, this is critical for combining a DPG method with transmission conditions as in Führer et al. (2017a) or with contact conditions as we show here. The singularly perturbed case is more technical for two reasons. First, the DPG setting itself for the PDE is more complicated (we use a robust formulation with three field variables) and, second, the combination of PDE and Signorini conditions has to take into account the diffusion coefficient as scaling parameter. In that way we apply what we have learnt from the DPG scheme (Heuer & Karkulik, 2017) for reaction-dominated diffusion and from the DPG–BEM coupling (Führer & Heuer, 2016) for this case. To the best of our knowledge, this is the first mathematical analysis of a numerical scheme for a singularly perturbed contact problem. We will see that there are symmetric and nonsymmetric forms to include the Signorini boundary conditions. In the symmetric case, we are able to provide a simple a posteriori error estimate that is based on the DPG energy error. Let us also remark that we focus on ultra-weak variational formulations. This has the advantage that both the trace and the flux appear as independent unknowns. This allows for a symmetric formulation of the Signorini conditions, needed for our a posteriori error bound. Other variational formulations can be considered analogously and they give rise to nonsymmetric well-posed formulations. In those cases, however, we have no a posteriori error analysis. Let us also mention that our scheme and analysis of the scalar Signorini problem can be extended to variational inequalities of the second kind, e.g., including Coulomb friction, and to linear elasticity so that, indeed, the Signorini problem can be solved by our DPG scheme. In the following, we continue this introduction by presenting our model problem, by introducing an abstract formulation as a variational inequality of the first kind that is suitable for the DPG scheme and by giving a final overview of the remainder of this article. 1.1 Model problem Let $${\it{\Omega}}\subset{\mathbb{R}}^d$$ ($$d\in\{2,3\}$$) be a simply connected Lipschitz domain with boundary $${\it{\Gamma}}=\partial{\it{\Omega}}$$ and unit normal vector $${\boldsymbol{n}}$$ on $${\it{\Gamma}}$$ pointing towards $${\mathbb{R}}^d\setminus\overline{\it{\Omega}}$$. For $$f\in L^2({\it{\Omega}})$$ we consider the model problem \begin{align}-c{\it{\Delta}} u + u = f \quad\text{in }{\it{\Omega}}, \end{align} (1.1a) subject to the Signorini boundary conditions \begin{align}\label{eq:intro:bc}u \geq 0, \quad \frac{\partial u}{\partial {\boldsymbol{n}}}\geq 0, \quad u \frac{\partial u}{\partial {\boldsymbol{n}}} = 0 \quad\text{on}\quad{\it{\Gamma}}. \end{align} (1.1b) Initially, we will study the case of constant $$c=1$$. Later, we will illustrate the applicability of our DPG scheme to the singularly perturbed problem with constant $$c=\varepsilon$$, assuming $$\varepsilon$$ to be a small positive number. Of course, the restriction of $$u$$ to $${\it{\Gamma}}$$ is understood in the sense of the trace, and its normal derivative on $${\it{\Gamma}}$$ is defined by duality. It is well known that this Signorini problem admits a unique solution $$u\in K := \{ v\in H^1({\it{\Omega}}) \,:\, v \geq 0\ \text{on}\ {\it{\Gamma}} \}$$ (see, e.g., Glowinski et al., 1981; Glowinski, 2008). Moreover, $$u$$ can be characterized as the unique solution of the variational inequality of the first kind \begin{align}\label{eq:intro:weakform}a(u,v-u) \geq (\,f,v-u) \quad\text{for all } v\in K, \end{align} (1.2) with \begin{align*}a(u,v): = {\rm{ }}\left( {\nabla u,\,\nabla v} \right) + {\rm{ }}\left( {u,\,v} \right)\quad {\text{for all }}u, v \in {H^1}({\it{\Omega}} ). \end{align*} In fact, by choosing appropriate test functions $$v\in K$$ and integrating by parts, one finds that problem (1.2) is equivalent to $$u\in K$$, (1.1a) and \begin{align}\label{eq:intro:bcalt}\int_{\it{\Gamma}} \frac{\partial u}{\partial{\boldsymbol{n}}} (v-u) \,{\rm{d}}{\it{\Gamma}} \geq 0 \quad\text{for all } v\in K; \end{align} (1.3) see, e.g., Glowinski et al. (1981). The last relation is useful for establishing a DPG setting of the variational inequality problem. 1.2 Variational formulation Let us give a brief overview of our variational setting for the DPG method. In Section 2, we will consider a nonstandard variational form of (1.1a): \begin{align}\label{eq:intro:dpg}\text{find } {\boldsymbol{u}}\in U \text{ s.t.}\quad b({\boldsymbol{u}},{\boldsymbol{v}}) = L({\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}} \in V. \end{align} (1.4) Here, $$U$$ and $$V$$ are different Hilbert spaces and $$b(\cdot,\cdot):\;U\times V\to{\mathbb{R}}$$ is the bilinear form stemming from, in our case, an ultra-weak formulation. At this point, no boundary conditions are included, so that there is no unique solution to (1.4). Denoting by $$({\cdot},{\cdot})_V$$ the inner product in $$V$$, we define the trial-to-test operator$${\it{\Theta}}_\beta:\;U\to V$$ by \begin{align}\label{eq:dpg:deftttop}{\it{\Theta}}_\beta := \beta{\it{\Theta}} \quad\text{with}\quad({{\it{\Theta}}{\boldsymbol{u}}},{{\boldsymbol{v}}})_V := b({\boldsymbol{u}},{\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}}\in V. \end{align} (1.5) The parameter $$\beta>0$$ has to be selected. Using this operator, the discretization of (1.4) will be based on its equivalent variant with so-called optimal test functions: \begin{align}\label{eq:intro:dpg:theta}\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) = L({\it{\Theta}}_\beta{\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}} \in U. \end{align} (1.6) In our formulations, only one component of $${\boldsymbol{u}}\in U$$ corresponds to the original unknown $$u$$. Depending on the particular case, we have to define appropriate Dirichlet and Neumann trace operators $$\gamma_0$$ and $$\gamma_{\boldsymbol{n}}$$ acting on $$U$$. Then, it is left to add the boundary conditions (1.1b) in the form $$\gamma_0 {\boldsymbol{u}}\geq 0$$, $$\gamma_{\boldsymbol{n}} {\boldsymbol{u}} \geq 0$$ and $$\gamma_0{\boldsymbol{u}} \gamma_{\boldsymbol{n}} {\boldsymbol{u}} = 0$$. This transforms (1.6) into a variational inequality. Keeping in mind (1.3), we define the bilinear form \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}) := b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{u}}},{\gamma_0 {\boldsymbol{v}}}\rangle_{\it{\Gamma}} \quad\text{for all } {\boldsymbol{u}},{\boldsymbol{v}} \in U \end{align*} and consider the following formulation: find $${\boldsymbol{u}}\in K^0 := \{ {\boldsymbol{v}}\in U \,:\, \gamma_0{\boldsymbol{v}} \geq 0 \}$$ such that \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all } {\boldsymbol{v}}\in K^0. \end{align*} We will show that this problem is equivalent to (1.1). In particular, it has a unique solution. An intrinsic feature of ultra-weak formulations is that all boundary conditions are essential. Therefore, we can derive methods that use different convex sets. From (1.1b) we infer that \begin{align*}u\left(\psi- \frac{\partial u}{\partial {\boldsymbol{n}}}\right) \geq 0 \quad\text{on }{\it{\Gamma}}\quad \text{for all } \psi\in H^{-1/2}({\it{\Gamma}}) \text{ with } \psi\geq 0, \end{align*} giving rise to the following formulation: find $${\boldsymbol{u}}\in K^{{\boldsymbol{n}}} := \left\{{{\boldsymbol{v}}\in U}:{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}\geq 0} \right\}$$ such that \begin{align*}a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) := b({\boldsymbol{u}},{\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}},{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all } {\boldsymbol{v}}\in K^{\boldsymbol{n}}. \end{align*} Note that neither $$a^0(\cdot,\cdot)$$ nor $$a^{\boldsymbol{n}}(\cdot,\cdot)$$ is symmetric. However, a combination leads to a formulation with symmetric bilinear form as follows: find $${\boldsymbol{u}}\in K^s := \left\{{{\boldsymbol{v}}\in U} \,:\, {\gamma_0{\boldsymbol{v}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{v}}\geq 0} \right\}$$ such that \begin{align*}a^s({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) := \tfrac12(a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}})+a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}})) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all }{\boldsymbol{v}}\in K^s. \end{align*} For this symmetric case we will establish a simple a posteriori error estimator. 1.3 Inhomogeneous boundary conditions Instead of (1.1b), the more general boundary conditions \begin{align*}u-g_D \geq 0, \quad \frac{\partial u}{\partial {\boldsymbol{n}}}-g_N\geq 0, \quad(u-g_D) \left(\frac{\partial u}{\partial {\boldsymbol{n}}} -g_N\right)= 0 \quad\text{on}\quad{\it{\Gamma}}, \end{align*} where $$g_{\rm D}$$ denotes the Dirichlet data and $$g_{\rm N}$$ the Neumann data, can be included in our variational setting as follows. As before one observes that \begin{align*}\int_{\it{\Gamma}} \left(\frac{\partial u}{\partial{\boldsymbol{n}}}-g_N\right) (v-u) \,{\rm{d}}{\it{\Gamma}} &\geq 0 \quad\text{for all } v\in H^1({\it{\Omega}}) \text{ with } v|_{\it{\Gamma}}\geq g_D. \end{align*} This gives rise to the following variational formulation: find $${\boldsymbol{u}}\in K^{g_D} := \left\{{{\boldsymbol{v}}\in U} \,:\, {\gamma_0{\boldsymbol{v}}\geq g_D} \right\}$$ s.t. \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) + \langle{g_N}, {\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\quad\text{for all }{\boldsymbol{v}}\in K^{g_D}. \end{align*} We note that the bilinear form $$a^0(\cdot,\cdot)$$ is used on the left-hand side. Similarly, one derives formulations using $$a^{\boldsymbol{n}}(\cdot,\cdot)$$, $$a^s(\cdot,\cdot)$$ with different convex sets. We stress that our analysis below holds true in these cases as well. Since, in general, $$K_h^\star\not\subseteq K^\star$$, the a priori analysis differs; see Remark 2.5. For the remainder of this work we stick to the boundary conditions (1.1b). 1.4 Overview The remainder of this article is structured as follows. Section 2 deals with the unperturbed model problem (1.1) (diffusion parameter $$c=1$$). After introducing some notation in Section 2.1, in Section 2.2 we present a variational inequality that represents (1.1) and is based on an ultra-weak variational formulation. Theorem 2.1 then states its well-posedness and equivalence. Afterwards, in Section 2.3 we present the discrete DPG approximation and prove its well-posedness (Theorem 2.2) and quasi-optimal convergence (Theorem 2.3). An a posteriori error estimate is derived in Section 2.4. Some technical results are collected in Section 2.5 and a proof of Theorem 2.1 is given at the end of Section 2. There is a similar structure in Section 3 for the singularly perturbed case (problem (1.1) with small diffusion $$c=\varepsilon$$). Notation is given in Section 3.1 and a variational formulation is presented and analysed in Section 3.2. There, Theorem 3.1 states the well-posedness and equivalence of the variational inequality. Section 3.3 presents and analyses the discrete scheme, its well-posedness (Theorem 3.2), quasi-optimality (Theorem 3.3) and an a posteriori error estimate (Theorem 3.4). Technical results and a proof of Theorem 3.1 are presented in Sections 3.4 and 3.5, respectively. Finally, in Section 4 we present several numerical examples that underline our theoretical results for the unperturbed and singularly perturbed cases. Throughout the article, $$a\lesssim b$$ means that $$a\le kb$$ with a generic constant $$k>0$$ that is independent of involved parameters, functions and the underlying mesh. Similarly, we use the notation $$a\simeq b$$. 2. DPG method 2.1 Notation For Lipschitz domains $$\omega\subset{\mathbb{R}}^d$$, we use the standard Sobolev spaces $$L^2(\omega)$$, $$H^1(\omega)$$ and $${\boldsymbol{H}}({{\rm div\,}},\omega)$$. The $$L^2(\omega)$$ resp. $$L^2(\partial\omega)$$ scalar products are denoted by $$(\cdot ,\cdot)_\omega$$ resp. $$\langle\cdot ,\cdot\rangle_{\partial\omega}$$ with induced norms $$\|\cdot\|_{\omega}$$ resp. $$\|\cdot\|_{{\partial\omega}}$$. Also, we define the trace spaces \begin{align*}H^{1/2}(\partial\omega) := \left\{ \gamma_\omega u\,:\, u\in H^1(\omega) \right\}\quad\text{ and its dual }\quad H^{-1/2}(\partial\omega) := \bigl(H^{1/2}(\partial\omega)\bigr)', \end{align*} where $$\gamma_\omega$$ denotes the trace operator. Here, duality is understood with respect to $$L^2(\partial\omega)$$ as a pivot space, i.e., using the extended $$L^2(\partial\omega)$$ inner product $$\langle{\cdot} ,{\cdot}\rangle_{\partial\omega}$$. The $$L^2({\it{\Omega}})$$ inner product will be denoted by $$({\cdot} ,{\cdot})$$ and the corresponding norm by $$\|\cdot\|_{}$$. Let $${\mathcal{T}}$$ denote a disjoint partition of $${\it{\Omega}}$$ into open Lipschitz sets $${T}\in{\mathcal{T}}$$, i.e., $$\bigcup_{{T}\in{\mathcal{T}}}\overline{T} = \overline{\it{\Omega}}$$. The set of all boundaries of all elements forms the skeleton $${\mathcal{S}} := \left\{ \partial{T} \mid {T}\in{\mathcal{T}} \right\}$$. By $${\boldsymbol{n}}_M$$ we mean the outer normal vector on $$\partial M$$ for a Lipschitz set $$M$$. On a partition $${\mathcal{T}}$$ we use the spaces \begin{align*}H^1({\mathcal{T}}) &:= \{ w\in L^2({\it{\Omega}}) \,:\, w|_T \in H^1(T) \,\forall\, T\in{\mathcal{T}}\}, \\ {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}}) &:= \{ {\boldsymbol{q}}\in (L^2({\it{\Omega}}))^d \,:\, {\boldsymbol{q}}|_T \in {\boldsymbol{H}}({{\rm div\,}},T) \,\forall\, T\in{\mathcal{T}}\}. \end{align*} The symbols $$\nabla_{\mathcal{T}}$$, $${\rm div}_{{\mathcal{T}}}\,$$ resp. $${\Delta}_{{\mathcal{T}}}\,$$ denote the $${\mathcal{T}}$$-piecewise gradient, divergence resp. Laplace operators. On the skeleton $${\mathcal{S}}$$ of $${\mathcal{T}}$$ we introduce the trace spaces \begin{align*}H^{1/2}({\mathcal{S}}) &:= \Big\{ \widehat u \in {\it{\Pi}}_{{T}\in{\mathcal{T}}}H^{1/2}(\partial{T})\,:\, \exists\, w\in H^1({\it{\Omega}}) \text{ such that }\widehat u|_{\partial{T}} = w|_{\partial{T}}\; \forall\, {T}\in{\mathcal{T}} \Big\},\\ H^{-1/2}({\mathcal{S}}) &:= \Big\{ \widehat\sigma \in {\it{\Pi}}_{{T}\in{\mathcal{T}}}H^{-1/2}(\partial{T})\,:\, \exists\, {\boldsymbol{q}}\in{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}) \text{ such that }\widehat\sigma|_{\partial{T}} = ({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \Big\}. \end{align*} These spaces are equipped with norms depending on the problem; see Sections 2.2 and 3.1. For functions $$\widehat u\in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma\in H^{-1/2}({\mathcal{S}})$$ and $${\boldsymbol\tau}\in{\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}})$$, $$v\in H^1({\mathcal{T}})$$ we define \begin{align*}\langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}}:= \sum_{{T}\in{\mathcal{T}}}\langle{\widehat u|_{\partial{T}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{T}}\rangle_{\partial{T}},\quad\langle{\widehat\sigma} ,{v}\rangle_{\mathcal{S}}:= \sum_{{T}\in{\mathcal{T}}}\langle{\widehat\sigma|_{\partial{T}}} ,{v}\rangle_{\partial{T}}. \end{align*} With the latter relations we can also define the restrictions $$\widehat u|_{\it{\Gamma}} \in H^{1/2}({\it{\Gamma}})$$ and $$\widehat \sigma|_{\it{\Gamma}} \in H^{-1/2}({\it{\Gamma}})$$ of functions $$\widehat u\in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma \in H^{-1/2}({\mathcal{S}})$$ onto $${\it{\Gamma}}$$. Let $$w\in H^1({\it{\Omega}})$$ be such that $$w|_{\partial T} = \widehat u|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. Then, \begin{align} \label{trace}\langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= \sum_{T\in{\mathcal{T}}}\langle{w|_{\partial T}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_T}\rangle_{\partial T} = \sum_{T\in{\mathcal{T}}} ({\nabla w} ,{{\boldsymbol\tau}})_T + ({w} ,{{{\rm div\,}} {\boldsymbol\tau}})_T = ({\nabla w} ,{{\boldsymbol\tau}}) + ({w} ,{{{\rm div\,}} {\boldsymbol\tau}}) \nonumber\\ &= \langle{w|_{\it{\Gamma}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}} =: \langle{\widehat u|_{\it{\Gamma}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}}\quad\text{for all }{\boldsymbol\tau}\in{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}). \end{align} (2.1) Note that $$\widehat u|_{\it{\Gamma}}$$ is uniquely determined since the above relation is independent of the choice of the extension $$w\in H^1({\it{\Omega}})$$ and since $$H^{-1/2}({\it{\Gamma}})$$ is the (normal) trace space of $${\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$. Similarly, we define $$\widehat\sigma|_{\it{\Gamma}} \in H^{-1/2}({\it{\Gamma}})$$ through \begin{align} \label{ntrace}\langle{\widehat \sigma} ,{v}\rangle_{\mathcal{S}} &= \sum_{T \in{\mathcal{T}}}\langle{{\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{T}} ,{v|_{\it{\Gamma}}}\rangle_{\partial T} = \sum_{T\in{\mathcal{T}}} ({{{\rm div\,}} {\boldsymbol\sigma}} ,{v})_T + ({{\boldsymbol\sigma}} ,{\nabla v})_T = ({{{\rm div\,}} {\boldsymbol\sigma}} ,{v}) + ({{\boldsymbol\sigma}} ,{\nabla v}) \nonumber\\ &= \langle{{\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{v|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} =: \langle{\widehat\sigma|_{\it{\Gamma}}} ,{v|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} \quad\text{for all } v\in H^1({\it{\Omega}}), \end{align} (2.2) where $${\boldsymbol\sigma}\in {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ with $${\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{T}|_{\partial T} = \widehat\sigma|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. 2.2 Ultra-weak variational formulation We derive an ultra-weak formulation of (1.1a) with $$c=1$$. Following Demkowicz & Gopalakrishnan (2011a), we define $${\boldsymbol\sigma} = \nabla u$$. Then, \begin{align*}-{{\rm div\,}}{\boldsymbol\sigma} + u = f, \quad {\boldsymbol\sigma} -\nabla u = 0. \end{align*} We define $$\widehat u \in H^{1/2}({\mathcal{S}})$$ and $$\widehat\sigma\in H^{-1/2}({\it{\Gamma}})$$ such that $$\widehat u|_{\partial T} = u|_{\partial T}$$ and $$\widehat\sigma|_{\partial T} = \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. Testing the first-order system with functions $$v\in H^1({\mathcal{T}})$$, $${\boldsymbol\tau}\in {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}})$$, and integrating by parts, we end up with the ultra-weak formulation \begin{align}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) + ({u} ,v) - \langle{\widehat\sigma} ,{v}\rangle_{{\mathcal{S}}} &= ({f} ,v), \\ \end{align} (2.3a) \begin{align}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{{\mathcal{S}}} &= 0. \end{align} (2.3b) This gives rise to a bilinear form $$b:U \times V \to {\mathbb{R}}$$ and functional $$L:V\to {\mathbb{R}}$$ defined by \begin{align*}b({\boldsymbol{u}},{\boldsymbol{v}}) &:= ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) + ({u} ,v) - \langle{\widehat\sigma} ,{v}\rangle_{{\mathcal{S}}}+ ({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{{\mathcal{S}}}, \\ L({\boldsymbol{v}}) &:= ({f} ,v) \end{align*} for all $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma) \in U$$, $${\boldsymbol{v}} = (v,{\boldsymbol\tau}) \in V$$, where \begin{align*}U &:= L^2({\it{\Omega}})\times [L^2({\it{\Omega}})]^d \times H^{1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}), \\ V &:= H^1({\mathcal{T}})\times {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}}). \end{align*} We equip these spaces with the norms \begin{align*}\|{{\boldsymbol{u}}}\|{_{U}^{2}} &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \|{\widehat u}\|_{1/2,{\mathcal{S}}}^2 +\|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}}^2, \\ \|{{\boldsymbol{v}}}\|_{V}^2 &:= \|{v}\|_{}^2 + \|{\nabla_{\mathcal{T}} v}\|_{}^2 + \|{{\boldsymbol\tau}}\|_{}^2 + \|{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}\|_{}^2, \end{align*} where the norms for the trace variables are given by the image norms (minimum energy extensions to $$H^1({\it{\Omega}})$$ resp. $${\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$) \begin{align*}\|{\widehat u}\|_{1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), \widehat u|_{\partial{T}}=w|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\|{{{\rm div\,}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}),\ \widehat\sigma|_{\partial{T}}\!=\!({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\!\in\!{\mathcal{T}} \right\}. \end{align*} In $$H^{1/2}({\it{\Gamma}})$$, we use the norm \begin{align*}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})} := \inf \left\{ (\|{w}\|_{}^2 + \|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}),\widehat v = w|_{\it{\Gamma}} \right\}, \end{align*} and define the norm $$\|{\cdot}\|_{H^{-1/2}({\it{\Gamma}})}$$ by duality. The bilinear form $$b(\cdot,\cdot)$$ induces a linear operator $$B:\;U\to V'$$ so that (2.3) can be written as \[\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad B{\boldsymbol{u}} = L. \] The operator $$B$$ has a nontrivial kernel. To consider the boundary conditions, we define trace operators \begin{align*}\begin{aligned}\gamma_0&:\;U\to H^{1/2}({\it{\Gamma}}),&\quad& \gamma_0(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma):=\widehat u|_{\it{\Gamma}}&\quad&\text{(Dirichlet trace, see (2.1)),} \\ \gamma_{\boldsymbol{n}}&:\;U\to H^{-1/2}({\it{\Gamma}}),&\quad& \gamma_{\boldsymbol{n}}(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma):=\widehat \sigma|_{\it{\Gamma}}&\quad&\text{(Neumann trace, see (2.2))}, \end{aligned}\end{align*} and define the sets \begin{align*}K^0 := \left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}} \geq 0} \right\}, \quad K^{\boldsymbol{n}} :=\left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_{\boldsymbol{n}}{\boldsymbol{u}}\geq 0} \right\}, \quad K^s := \left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{u}}\geq 0} \right\}. \end{align*} The relations ‘$$\geq$$’ are partial orderings. Following Kikuchi & Oden (1988, Section 5) we define \begin{align*}\widehat v \geq 0 \text{ in } H^{1/2}({\it{\Gamma}}) \quad :\Longleftrightarrow \quad\exists\, \{\widehat v_n\}_{n\in {\mathbb{N}}} \text{ s.t. } v_n \in \mathrm{Lip}({\it{\Gamma}}) \text{ with }v_n\geq 0 \text{ and } v_n \rightharpoonup v \quad\text{in } H^{1/2}({\it{\Gamma}}), \end{align*} where $$\mathrm{Lip}({\it{\Gamma}})$$ denotes all Lipschitz continuous functions $${\it{\Gamma}} \to {\mathbb{R}}$$. We note that this definition of $$\widehat v\geq 0$$ is equivalent to $$\widehat v\geq 0$$ a.e. (see Kikuchi & Oden, 1988, Section 5). On $$H^{-1/2}({\it{\Gamma}})$$, ‘$$\geq$$’ is understood as duality (see, e.g., Hlaváček et al., 1988, Section 1.1.11), \begin{align*}\widehat\sigma \geq 0 \text{ in } H^{-1/2}({\it{\Gamma}}) \quad:\Longleftrightarrow \quad\langle{\widehat\sigma} ,{\widehat v}\rangle_{\it{\Gamma}} \geq 0 \quad\text{for all } \widehat v\in H^{1/2}({\it{\Gamma}}) \text{ with } \widehat v\geq 0. \end{align*} One can verify that $$\left\{{\widehat v\in H^{1/2}({\it{\Gamma}})} \,:\, {\widehat v\geq 0} \right\}$$ and $$\left\{{\widehat\sigma\in H^{-1/2}({\it{\Gamma}})} \,:\, {\widehat\sigma\geq 0} \right\}$$ are closed, convex sets. We will also see that $$K^0$$, $$K^{\boldsymbol{n}}$$ and $$K^s$$ are nonempty, closed, convex subsets of $$U$$ (see Lemma 2.8 below). Now let $$u\in H^1({\it{\Omega}})$$ denote the solution of problem (1.1) and define $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in U$$ with components as above. Then $$B{\boldsymbol{u}}=L$$ and, considering inequality (1.3) as a representation of the boundary conditions (1.1b), one sees that $${\boldsymbol{u}}\in K^\star$$ for $$\star\in\{0,{\boldsymbol{n}},s\}$$ and \begin{align*}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\geq 0 \quad\text{for all }{\boldsymbol{v}}\in K^0, \\ \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\geq 0 \quad\text{for all } {\boldsymbol{v}}\in K^{\boldsymbol{n}}, \\ \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}) &\geq 0 \quad\text{for all } {\boldsymbol{v}}\in K^s. \end{align*} Recalling the formulation (1.6) of the problem $$B{\boldsymbol{u}}=L$$, this leads us to defining the bilinear forms $$a^\star:U\times U\to {\mathbb{R}}$$, $$\star\in\{0,{\boldsymbol{n}},s\}$$ and the functional $$F:U\to{\mathbb{R}}$$ as \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}, \\ a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}, \\ a^s({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}), \\ F({\boldsymbol{v}}) &:= L({\it{\Theta}}_\beta{\boldsymbol{v}}) \end{align*} for $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. Here, $$\beta>0$$ is a constant that will be fixed below. We then obtain the following formulations of the model problem (1.1) (with $$c=1$$) as ultra-weak variational inequalities: for fixed $$\star\in\{0,{\boldsymbol{n}},s\}$$ find $${\boldsymbol{u}}\in K^\star$$ such that \begin{align}\label{eq:dpg:varineq}a^\star({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq F({\boldsymbol{v}}-{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{v}}\in K^\star. \end{align} (2.4) These are variational inequalities of the first kind, and we use a standard framework for their analysis (cf. Glowinski et al., 1981; Glowinski, 2008). The following is one of our main results. Theorem 2.1 Fix $$\star\in\{0,{\boldsymbol{n}},s\}$$. For all $$\beta\geq 2$$ the following holds: the bilinear form $$a^\star:U\times U \to {\mathbb{R}}$$ is $$U$$-coercive, \begin{align*}\|{{\boldsymbol{u}}}\|_{U}^2 \leq C_1 a^\star({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{u}}\in U, \end{align*} and bounded, \begin{align*}|a^\star({\boldsymbol{u}},{\boldsymbol{v}})| \leq (C_2^2 \beta+1) \|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U} \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}} \in U. \end{align*} The constant $$C_1>0$$ depends only on $${\it{\Omega}}$$ and $$C_2=\|{b}\|_{}=\|{B}\|_{}$$. In particular, the variational inequality (2.4) is uniquely solvable and equivalent to problem (1.1) with $$c=1$$ in the following sense. If $$u\in H^1({\it{\Omega}})$$ solves problem (1.1), then $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in K^\star$$ with $${\boldsymbol\sigma} := \nabla u$$, $$\widehat u|_{\partial T} := u|_{\partial T}$$, $$\widehat\sigma|_{\partial T} := \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$ solves (2.4). On the other hand, if $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in K^\star$$ solves (2.4), then $$u\in H^1({\it{\Omega}})$$ solves (1.1). Moreover, the unique solution $${\boldsymbol{u}}\in K^\star$$ of (2.4) satisfies \begin{align}\label{eq:dpg:exactsol}b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{w}}) = F({\boldsymbol{w}}) \quad\text{for all } {\boldsymbol{w}}\in U. \end{align} (2.5) This theorem is proved in Section 2.6. 2.3 Discretization and convergence To discretize our variational inequality (2.4), we use, in this work, lowest-order piecewise polynomial functions. That is, we replace the space $$U$$ by \begin{align*}U_h := P^0({\mathcal{T}})\times [P^0({\mathcal{T}})]^d \times S^1({\mathcal{S}}) \times P^0({\mathcal{S}}), \end{align*} where $$P^0$$ denotes the space of elementwise constants on $${\mathcal{T}}$$ resp. $${\mathcal{S}}$$, and $$S^1({\mathcal{S}})$$ is the space of globally continuous, elementwise affine functions on $${\mathcal{S}}$$. Defining the nonempty convex subsets \begin{align*}K_h^0 &:= \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0} \right\}, \\ K_h^{\boldsymbol{n}} &:= \left\{{{\boldsymbol{v}}_h\in U_h}:{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}_h\geq 0} \right\}, \\ K_h^s &:= \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h\geq 0} \right\}, \end{align*} we find that $$K_h^\star\subseteq K^\star$$. The discretized version of (2.4) then reads, for fixed $$\star\in\{0,{\boldsymbol{n}},s\}$$ find $${\boldsymbol{u}}_h\in K_h^\star$$ such that \begin{align}\label{eq:dpg:varineqdisc}a^\star({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h)\geq F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \quad\text{for all } {\boldsymbol{v}}_h\in K_h^\star. \end{align} (2.6) Coercivity and boundedness of $$a^\star(\cdot,\cdot)$$ hold on the full space $$U$$ (Theorem 2.1). Therefore, the Lions–Stampacchia theorem applies also in the discrete case. Theorem 2.2 Under the assumptions of Theorem 2.1, the discrete variational inequality (2.6) admits a unique solution $${\boldsymbol{u}}_h\in K_h^\star$$. Our scheme converges quasi-optimally in the following sense. Theorem 2.3 For $$\beta\ge 2$$ let $${\boldsymbol{u}}\in K^\star$$, $${\boldsymbol{u}}_h\in K_h^\star$$ denote the exact solutions of (2.4), (2.6). Then it holds that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \begin{cases}\inf_{{\boldsymbol{v}}_h\in K_h^0} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 +\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}\right) & (\star = 0), \\ \inf_{{\boldsymbol{v}}_h\in K_h^{\boldsymbol{n}}} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 +\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\right) & (\star = {\boldsymbol{n}}), \\ \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 + \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right) & (\star = s). \end{cases}\end{align*} The generic constant depends on $${\it{\Omega}}$$ and $$\beta$$ but not on $${\mathcal{T}}$$. Proof. By Theorem 2.1, $$a^\star(\cdot,\cdot)$$ is $$U$$-coercive and bounded. Therefore, we can follow Falk’s lemma (Falk, 1974) to deduce that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2 &\lesssim a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{u}}_h) = a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{v}}_h) + a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ & \leq \|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U (C_2^2\beta+1) \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U + a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \end{align*} for all $${\boldsymbol{v}}_h\in K_h^\star\subseteq K^\star$$. We consider only the case $$\star = 0$$. The remaining cases are treated in the same manner. To tackle the last term on the right-hand side, we use (2.5) and (2.6) to see that \begin{align*}a^0({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) &= b({\boldsymbol{u}},{\it{\Theta}}_\beta ({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) ) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}- a^0({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ &= \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}} + F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)-a^0({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ & \leq \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}. \end{align*} Note that the exact solution $${\boldsymbol{u}}$$ satisfies $$\gamma_{\boldsymbol{n}} {\boldsymbol{u}} \geq 0$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = 0$$. For the discrete one it holds that $$\gamma_0{\boldsymbol{u}}_h\geq 0$$. Hence, the last term further simplifies to \begin{align*}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}} \leq \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0 {\boldsymbol{v}}_h}\rangle_{\it{\Gamma}}= \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}. \end{align*} Altogether, Young’s inequality with some parameter $$\delta>0$$ shows that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2\lesssim \frac{\delta}2 \|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2 + \frac{\delta^{-1}}2 (C_2^2\beta+1)^2\|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}\end{align*} for arbitrary $${\boldsymbol{v}}_h\in K_h^0$$. This proves the a priori estimate. □ Let $${\mathcal{S}}_{\it{\Gamma}} := \left\{{\partial T \cap {\it{\Gamma}}} \,:\, {T\in{\mathcal{T}}} \right\}$$ denote the mesh on the boundary, which is induced by the volume mesh $${\mathcal{T}}$$. Remark 2.4 To deduce convergence rates assume, for instance, that $$u\in H^3({\it{\Omega}})$$ is the solution of (1.1). Set \begin{align*}{\boldsymbol{v}}_h := ({\it{\Pi}}_h^0 u,\boldsymbol{{\it{\Pi}}}_h^0\nabla u,I_h u|_{\mathcal{S}}, ({\it{\Pi}}_h^{{{\rm div\,}}} \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T})_{T\in{\mathcal{T}}}) \in U, \end{align*} where $${\it{\Pi}}_h^0$$ and $$\boldsymbol{{\it{\Pi}}}_h^0$$ are the $$L^2({\it{\Omega}})$$-orthogonal projections onto the elementwise constant spaces, $$I_h$$ is the nodal interpolant and $${\it{\Pi}}_h^{{{\rm div\,}}}$$ is the (lowest-order) Raviart–Thomas projector. Note that $$I_h$$ preserves non-negativity (in particular, on the boundary) and the normal trace of $${\it{\Pi}}_h^{{{\rm div\,}}}$$ is the $$L^2({\it{\Gamma}})$$-orthogonal projection $$\pi_h^0$$ and, hence, preserves non-negativity on the boundary as well. Thus, $${\boldsymbol{v}}_h \in K_h^\star$$ for $$\star\in\{0,{\boldsymbol{n}},s\}$$. We refer to Demkowicz & Gopalakrishnan (2011a) to see that $$\|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U = {\mathcal O}(h)$$. To control the boundary terms, we note that \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}_h-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| \leq \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{{\it{\Gamma}}}\|{(u-I_h u)|_{\it{\Gamma}}}\|_{{\it{\Gamma}}} = {\mathcal{O}}(h^2). \end{align*} Let $${\it{\Gamma}}_1,\dots,{\it{\Gamma}}_L \subseteq {\it{\Gamma}}$$ be such that $$\bigcup \overline{\it{\Gamma}}_j = {\it{\Gamma}}$$ and $${\boldsymbol{n}}_{\it{\Omega}}|_{{\it{\Gamma}}_j}$$ is constant. Using the projection properties of $${\it{\Pi}}_h^{{{\rm div\,}}}$$ and $$\pi_h^0$$ and the local approximation property of $$\pi_h^0$$ we obtain \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| &\leq \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_E| = \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}-\pi_h^0\gamma_0{\boldsymbol{u}}}\rangle_E| \\ & \lesssim \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |E| \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^1(E)} |E| \|{\gamma_0{\boldsymbol{u}}}\|_{H^1(E)}\lesssim h^2 \left(\sum_{j=1}^L \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^1({\it{\Gamma}}_j)}^2\right)^{1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^1({\it{\Gamma}})}. \end{align*} Altogether, using $${\boldsymbol{v}}_h$$ in Theorem 2.3, we infer \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 &\lesssim \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 + \begin{cases}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}| & (\star = 0) \\ |\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| & (\star = {\boldsymbol{n}}) \\ \tfrac12 |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| & (\star = s) \end{cases}\\&= {\mathcal O}(h^2). \end{align*} With less regularity of $$u$$ the treatment of the boundary terms becomes more technical. We refer to Drouet & Hild (2015) for details. Remark 2.5 Our analysis also allows for nonconforming discrete cones $$K_h^\star\not\subseteq K^\star$$. Then, an additional consistency error shows up in Theorem 2.3, i.e., in the case $$\star = 0$$, \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^0} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}_h-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \right) + \inf_{{\boldsymbol{v}}\in K^0}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}. \end{align*} Again, this a priori estimate can be derived using Falk’s lemma. 2.4 A posteriori error estimate We derive a simple error estimator in the symmetric case, i.e., $$\star=s$$. Throughout this section we assume that $$\beta>0$$ is a fixed constant, such that $$a^s(\cdot,\cdot)$$ is coercive (Theorem 2.1). Let $${\boldsymbol{u}}_h\in K_h^s$$ denote the unique solution of (2.6). We define for all $$T\in{\mathcal{T}}$$, local volume error indicators $$\eta(T)$$ as the norm of the residual $$L-B{\boldsymbol{u}}_h$$ restricted to $$T$$, or, formally, \begin{align} \label{eq:aposteriori:estdef}\eta(T)^2 := \beta \|{R_T^{-1}\iota_T^*(L-B{\boldsymbol{u}}_h)}\|_{V(T)}^2. \end{align} (2.7) Here, $$V(T) := H^1(T) \times {\boldsymbol{H}}({{\rm div\,}},T)$$, $$\|{\cdot}\|_{V(T)}$$ denotes the canonical norm on $$V(T)$$, $$R_T:\;V(T) \to (V(T))'$$ is the Riesz isomorphism, and $$\iota_T^*$$ is the dual of the canonical embedding $$\iota_T:\; V(T)\to V$$. Moreover, for all $$E\in{\mathcal{S}}_{\it{\Gamma}}$$ we define local boundary indicators by \begin{align*}\eta(E)^2 := \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_E. \end{align*} Note that $${\boldsymbol{u}}_h\in K_h^s$$ implies that $$\eta(E)^2\geq 0$$. The overall estimator is then given by \begin{align*}\eta^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} Theorem 2.6 For $$\beta\ge 2$$ let $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ be the solutions of (2.4) and (2.6), respectively. Then, there holds the reliability estimate \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U \leq C_\mathrm{rel} \eta, \end{align*} with a constant $$C_\mathrm{rel}>0$$ that depends on $${\it{\Omega}}$$ but not on $${\mathcal{T}}$$ or $$\beta$$. Proof. By the $$U$$-coercivity of $$a^s(\cdot,\cdot)$$ (see Theorem 2.1) we have \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 &\lesssim a^s({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{u}}_h) = b({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\it{\Theta}}_\beta({\boldsymbol{u}}-{\boldsymbol{u}}_h) ) + \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-{\boldsymbol{u}}_h)} ,{\gamma_0({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}\\ &=\beta \|{B({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-{\boldsymbol{u}}_h)} ,{\gamma_0({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}. \end{align*} Note that $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ imply that $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}\geq 0$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\geq 0$$. Together with $$B{\boldsymbol{u}} = L$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = 0$$ we obtain the estimate \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \beta \|{L-B{\boldsymbol{u}}_h}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}= \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2, \end{align*} which finishes the proof. □ 2.5 Technical details We start by proving boundedness of our trace operators that are specifically modified for the space $$U$$. Lemma 2.7 The operators $$\gamma_0 : (U,\|\cdot\|_{U})\to (H^{1/2}({\it{\Gamma}}),\|\cdot\|_{H^{1/2}({\it{\Gamma}})})$$ and $$\gamma_{\boldsymbol{n}} : (U,\|\cdot\|_{U})\to (H^{-1/2}({\it{\Gamma}}), \|\cdot\|_{H^{-1/2}({\it{\Gamma}})})$$ have unit norm. Proof. Boundedness of these operators follows basically by definition of the corresponding norms (see also Führer et al., 2017a, Lemma 3). Specifically, the definitions of the norms $$\|\cdot\|_{H^{1/2}({\it{\Gamma}})}$$, $$\|\cdot\|_{1/2,{\mathcal{S}}}$$ and $$\|\cdot\|_{U}$$ prove boundedness of $$\gamma_0$$ with bound $$1$$. That the norm of these operators is equal to $$1$$ follows by considering appropriate extensions. Note that $$\|{\cdot|_{\it{\Gamma}}}\|_{H^{-1/2}({\it{\Gamma}})} \leq \|{\cdot}\|_{-1/2,{\mathcal{S}}}$$ and, thus, the definition of $$\|\cdot\|_{U}$$ shows boundedness of $$\gamma_{\boldsymbol{n}}$$. □ By the boundedness of the operators $$\gamma_0$$ and $$\gamma_{\boldsymbol{n}}$$ we immediately establish the following result. Lemma 2.8 The sets $$K^\star$$ ($$\star\in\{0,{\boldsymbol{n}},s\}$$) are nonempty, closed, convex subsets of $$U$$. The following steps are to characterize the kernel of the operator $$B$$. This kernel is nontrivial since $$B$$ does not include any boundary condition. Our procedure is similar to the one in Führer et al. (2017a). For a function $$v\in H^{1/2}({\it{\Gamma}})$$, we define its quasi-harmonic extension $$\widetilde u\in H^1({\it{\Omega}})$$ as the unique solution of \begin{align}\label{eq:dpg:harmext}-{\it{\Delta}} \widetilde u + \widetilde u &=0 \quad\text{in }{\it{\Omega}}, \quad\widetilde u|_{\it{\Gamma}} = v. \end{align} (2.8) Note that the infimum in the definition of $$\|{v}\|_{H^{1/2}({\it{\Gamma}})}$$ is attained for the function $$\widetilde u$$, i.e., $$\|{\widetilde u}\|_{}^2 + \|{\nabla\widetilde u}\|_{}^2 = \|{v}\|_{H^{1/2}({\it{\Gamma}})}^2$$. Then, define the operator $$\mathcal{E} : H^{1/2}({\it{\Gamma}})\to U$$ by \begin{align*}\mathcal{E} v := (\widetilde u,{\boldsymbol\sigma},\widehat u,\widehat\sigma), \text{ where }{\boldsymbol\sigma} := \nabla\widetilde u, \quad \widehat u|_{\partial T} := \widetilde u|_{\partial T}, \quad \widehat\sigma|_{\partial T} := \nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T} \quad\text{for all } T\in{\mathcal{T}}. \end{align*} We combine important properties of $$B$$ and $$\mathcal{E}$$ in the following lemma. Lemma 2.9 The operators $$B : U \to V'$$, $$\mathcal{E}: H^{1/2}({\it{\Gamma}})\to U$$ have the following properties: (i) The operators $$B$$ and $$\mathcal{E}$$ are bounded. Specifically it holds that \begin{align*}&\|{B{\boldsymbol{u}}}\|_{V'} \lesssim \|{{\boldsymbol{u}}}\|_{U} &&\quad\text{for all } {\boldsymbol{u}}\in U,\\ &\|{v}\|_{H^{1/2}({\it{\Gamma}})} \le \|{\mathcal{E} v}\|_{U} \le \sqrt{3} \|{v}\|_{H^{1/2}({\it{\Gamma}})}&&\quad\text{for all } v \in H^{1/2}({\it{\Gamma}}). \end{align*} (ii) The kernel of $$B$$ consists of all quasi-harmonic extensions, i.e., $$\ker(B) = \operatorname{ran}(\mathcal{E})$$. (iii) The operator $$\mathcal{E}$$ is a right inverse of $$\gamma_0$$. (iv) The operator $$B:\;U/\ker(B)\to V'$$ is inf–sup stable, \begin{align*}\|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U} \lesssim \|{B{\boldsymbol{u}}}\|_{V'} = \sup_{{\boldsymbol{v}}\in V} \frac{\langle{B{\boldsymbol{u}}} ,{{\boldsymbol{v}}}\rangle}{\|{{\boldsymbol{v}}}\|_{V}}= b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align*} The involved constant depends only on $${\it{\Omega}}$$. Proof. In the case of the Poisson equation, these results have been established in Führer et al. (2017a). In a recent work (Führer et al., 2017b), we analysed a time-stepping scheme for the heat equation, which naturally leads to the equation $$-{\it{\Delta}} u + \delta^{-1}u = f$$, where $$\delta$$ corresponds to a time step $$k_n$$. Setting $$\delta=k_n=1$$ in Führer et al. (2017b, Lemma 8), we obtain boundedness of the operator $$B$$ and stability \begin{align*}\|{{\boldsymbol{u}}_0}\|_{U} \lesssim \|{B{\boldsymbol{u}}_0}\|_{V'} \quad\text{for all } {\boldsymbol{u}}_0 \in U \text{ with } \gamma_0{\boldsymbol{u}}_0 = 0. \end{align*} By definition of $$\mathcal{E}$$ one sees (iii). Furthermore, integration by parts shows $$\operatorname{ran}(\mathcal{E})\subseteq \ker(B)$$. For $${\boldsymbol{u}}\in U$$ we define $${\boldsymbol{u}}_0:={\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}$$ and infer \begin{align*}\|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_U \lesssim \|{B({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})}\|_{V'} = \|{B{\boldsymbol{u}}}\|_{V'}, \end{align*} which proves (iv) as well as $$\ker(B)\subseteq \operatorname{ran}(\mathcal{E})$$, hence (ii). It remains to show the relations for $$\mathcal{E}$$ in (i). Let $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma) = \mathcal{E} v$$ for $$v\in H^{1/2}({\it{\Gamma}})$$. Then, \begin{align*}\|{\mathcal{E} v}\|_U^2 \geq \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 = \|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 = \|{v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} On the other hand, using that $${\it{\Delta}} u = u$$ by construction of $$\mathcal{E} v$$, we deduce that \begin{align*}\|{\mathcal{E} v}\|_U^2 &= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \|{\widehat u}\|_{1/2,{\mathcal{S}}}^2 + \|{\widehat\sigma}\|_{-1/2,{\it{\Gamma}}}^2 \\ &\leq \|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 +\|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 + \|{\nabla u}\|_{}^2 + \|{{\it{\Delta}} u}\|_{}^2 \\ &= 3\|{u}\|_{}^2 + 3\|{\nabla u}\|_{}^2 = 3\|{v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} This concludes the proof. □ In Lemma 2.11 below, we give an explicit bound for the control of the Neumann trace for functions of the quotient space $$U/\ker(B)$$. For its proof we need the following technical result. Lemma 2.10 Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$. The problem \begin{align}{{\rm div\,}}{\boldsymbol\tau} + v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:dpg:dualestimate:a} \\ \end{align} (2.9a) \begin{align}{\boldsymbol\tau} + \nabla v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:dpg:dualestimate:b} \\ \end{align} (2.9b) \begin{align}v|_{{\it{\Gamma}}} &= \widehat v \label{eq:dpg:dualestimate:c}\end{align} (2.9c) admits a unique solution $$(v,{\boldsymbol\tau}) \in H^1({\it{\Omega}})\times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ with $${\it{\Delta}} v\in H^1({\it{\Omega}})$$ and \begin{align*}\|{{\boldsymbol\tau}}\|_{}^2 + \|{{{\rm div\,}}{\boldsymbol\tau}}\|_{}^2 = \|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 = \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Proof. Let $$v\in H^1({\it{\Omega}})$$ be the unique solution of \begin{align*}-{\it{\Delta}} v + v = 0\quad\text{in } {\it{\Omega}}, \quad v|_{\it{\Gamma}} = \widehat v. \end{align*} Then, $$\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 = \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2$$ by definition of the latter norm. Define $${\boldsymbol\tau}:=-\nabla v\in L^2({\it{\Omega}})$$. Since $${\it{\Delta}} v = v \in H^1({\it{\Omega}})$$, we have $${\boldsymbol\tau}\in {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$, and $${{\rm div\,}}{\boldsymbol\tau} = -{\it{\Delta}} v = -v$$ shows (2.9a). To see unique solvability, let additionally $$(v_2,{\boldsymbol\tau}_2)\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ solve (2.9). The difference $$w:=v-v_2$$ satisfies $$-{\it{\Delta}} w + w =0$$ in $${\it{\Omega}}$$ with $$w|_{\it{\Gamma}} = 0$$. Thus, $$w=0$$ and $${\boldsymbol\tau} = -\nabla v = -\nabla v_2 = {\boldsymbol\tau}_2$$ as well. □ Lemma 2.11 It holds that \begin{align}\label{eq:dpg:Hm12estimate}\|{\gamma_{\boldsymbol{n}}\|({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})}\|_{H^{-1/2}({\it{\Gamma}})} \leq \sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align} (2.10) Proof. Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ and choose the test function $${\boldsymbol{v}}=(v,{\boldsymbol\tau})\in H^1({\it{\Omega}}) \times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}) \subseteq V$$ to be the solution of (2.9). By Lemma 2.10 it holds $$\|{{\boldsymbol{v}}}\|_{V}=\sqrt{2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}$$. Then, by the definition of the bilinear form $$b(\cdot,\cdot)$$ and (2.1), (2.2) we have $$b({\boldsymbol{u}},{\boldsymbol{v}}) = -\langle{\widehat\sigma} ,{v}\rangle_{\mathcal{S}} -\langle{\widehat u} ,{\tau\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} = -\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\widehat v}\rangle_{\it{\Gamma}} - \langle{\gamma_0{\boldsymbol{u}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}}$$. Since $$\gamma_0({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}) = 0$$ it follows that \begin{align*}\lvert \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\widehat v}\rangle_{\it{\Gamma}} \rvert &= \lvert b({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}},{\boldsymbol{v}})\rvert = \lvert b({\boldsymbol{u}},{\boldsymbol{v}})\rvert \leq \|{B{\boldsymbol{u}}}\|_{V'} \|{{\boldsymbol{v}}}\|_{V}= \sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}, \end{align*} where we have used that $$\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}\in \ker(B)$$. Dividing by $$\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}$$ and taking the supremum over all $$\widehat v\in H^{1/2}({\it{\Gamma}}) \setminus \{0\}$$, this proves (2.10). □ 2.6 Proof of Theorem 2.1 First, we prove boundedness and coercivity of $$a^\star(\cdot,\cdot)$$. Then, the Lions–Stampacchia theorem (see, e.g., Glowinski et al., 1981; Rodrigues, 1987; Glowinski, 2008) proves unique solvability of (2.4) provided that $$F : U\to {\mathbb{R}}$$ is a linear functional, which follows by the boundedness of $${\it{\Theta}}:U\to{\mathbb{R}}$$. We show the boundedness. Note that $$b:U\times V\to {\mathbb{R}}$$ and $${\it{\Theta}}:U\to V$$ are uniformly bounded, \begin{align*}|b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}})| \leq \beta C_2 \|{{\boldsymbol{u}}}\|_{U}\|{{\it{\Theta}}{\boldsymbol{v}}}\|_{V} \leq \beta C_2^2\|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U}\quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}}\in U. \end{align*} By duality and boundedness of $$\gamma_0,\gamma_{\boldsymbol{n}}$$ (see Lemma 2.7) we have \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^{-1/2}({\it{\Gamma}})}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})} \leq \|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U} \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}}\in U. \end{align*} Next, we follow Führer et al. (2017a) to prove coercivity. Note that $$a^0({\boldsymbol{u}},{\boldsymbol{u}}) = a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{u}}) = a^s({\boldsymbol{u}},{\boldsymbol{u}})$$ for all $${\boldsymbol{u}}\in U$$. Let $${\boldsymbol{u}}\in U$$. By the triangle inequality and Lemma 2.9 we get \begin{align*}\|{\boldsymbol{u}}\|_{U}^2 \lesssim \|{{\boldsymbol{u}}-{\mathcal{E}}\gamma_0{\boldsymbol{u}}}\|_U^2 + \|{{\mathcal{E}}\gamma_0{\boldsymbol{u}}}\|_{U}^2 \lesssim \|{B{\boldsymbol{u}}}\|_{V'}^2 + \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Let $$\widetilde u\in H^1({\it{\Omega}})$$ be the quasi-harmonic extension (2.8) of $$\gamma_0{\boldsymbol{u}}$$. The definition of the $$H^{1/2}({\it{\Gamma}})$$-norm and integration by parts show that \begin{align*}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 &= \|{\widetilde u}\|_{H^1({\it{\Omega}})}^2 = \|{\widetilde u}\|_{L^2({\it{\Omega}})}^2 + \|{\nabla \widetilde u}\|_{L^2({\it{\Omega}})}^2 = \langle{\partial_{\boldsymbol{n}} \widetilde u} ,{\widetilde u|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} = \langle{\gamma_{\boldsymbol{n}} \mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} This gives \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} Using duality, Lemma 2.11 and Young’s inequality, we obtain \begin{align*}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\leq \|{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^{-1/2}({\it{\Gamma}})} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}\leq \sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \\ &\leq \|{B{\boldsymbol{u}}}\|_{V'}^2 + \frac{1}2 \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Altogether, this shows \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \tfrac12 \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \leq 2 \|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} Thus, \begin{align*}C_1^{-1} \|{{\boldsymbol{u}}}\|_{U^2} \leq \beta\|{B{\boldsymbol{u}}}\|_{V'}^2 +\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = a^\star({\boldsymbol{u}},{\boldsymbol{u}}) \end{align*} for all $$\beta\geq 2$$, where the constant $$C_1>0$$ depends only on $${\it{\Omega}}$$. Regarding the equivalence of problems (1.1) and (2.4) we know that the unique solution $$u$$ of (1.1) with $${\boldsymbol{u}}$$ defined as in the assertion satisfies (2.4) by construction. The other direction follows by existence of a unique solution of (2.4). Finally, note that (2.5) follows by construction. However, in the case $$\star=0$$ we can infer this identity directly from (2.4) as follows. Let $${\boldsymbol{u}}$$ denote the solution of (2.4) with $$\star=0$$. Set $${\boldsymbol{v}} = {\boldsymbol{u}}\pm ({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})$$ for some arbitrary $${\boldsymbol{w}}\in U$$. Since $$\gamma_0({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) = 0$$, we infer $${\boldsymbol{v}}\in K^0$$, so that we can use it as a test function in (2.4). This gives \begin{align*}\pm a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) \geq \pm F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}). \end{align*} Hence, \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) = F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) \quad\text{for all }{\boldsymbol{w}}\in U. \end{align*} Note that $$\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}\in\ker B = \ker {\it{\Theta}}$$. This leads to \begin{align*}F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) &= L({\it{\Theta}}_\beta ({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})) = L({\it{\Theta}}_\beta{\boldsymbol{w}}) = F({\boldsymbol{w}}) \text{ and }\\ a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) &= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{w}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})}\rangle_{\it{\Gamma}}= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{w}}). \end{align*} This concludes the proof. 3. DPG method for a singularly perturbed problem In this section, we introduce and analyse a DPG method for the Signorini problem of reaction-dominated diffusion, i.e., for (1.1) with small positive constant $$c=\varepsilon$$ ($$0<\varepsilon\le 1$$). We stick mainly to the notation as given in Section 2 but need some additional definitions. Also, we redefine some objects like the bilinear form $$b:\;U\times V\to{\mathbb{R}}$$, the spaces $$U$$, $$V$$ and some norms. In fact, our objective of robust control of field variables forces us to carefully scale parts of norms with coefficients depending on the diffusion parameter $$\varepsilon$$. The ultra-weak formulation taken from Heuer & Karkulik (2017) is derived by rewriting problem (1.1) (with $$c=\varepsilon$$) as a first-order system \begin{align*}\rho-{\mathrm{div}}\,{\boldsymbol\sigma} = 0, \qquad \varepsilon^{-1/4}{\boldsymbol\sigma} - \nabla u = 0, \qquad -\varepsilon^{3/4}\rho + u = f. \end{align*} We test the first two equations with $$\mu\in H^1({\mathcal{T}})$$ resp. $${\boldsymbol\tau}\in {\boldsymbol{H}}({\mathrm{div}},{\mathcal{T}})$$ elementwise, integrate by parts and sum over all elements. The third equation is tested with $$v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v$$ for $$v\in H^1({\it{\Delta}},{\mathcal{T}})$$. Integrating by parts and using the first two equations, we get the formulation \begin{align}({\rho} ,{\mu}) + ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}}\mu}) - \langle{\widehat \sigma^a} ,{\mu}\rangle_{\mathcal{S}} &= 0, \\ \end{align} (3.1a) \begin{align}\varepsilon^{-1/4}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u^a} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= 0, \\ \end{align} (3.1b) \begin{align}\begin{split}\varepsilon^{3/4}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{3/4} \langle{\widehat\sigma^b} ,{v}\rangle_{\mathcal{S}} + ({u} ,{v}) \qquad\qquad\qquad\qquad& \quad\\ +{}\varepsilon^{5/4} ({\rho} ,{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) + \varepsilon^{1/4} ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{1/2} \langle{\widehat u^b} ,{\nabla_{\mathcal{T}}v\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= ({f} ,{v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v}). \end{split}\end{align} (3.1c) 3.1 Notation For functions $$\widehat u \in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma\in H^{-1/2}({\mathcal{S}})$$ we define the skeleton norms \begin{align*}\|{\widehat u}\|_{1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \varepsilon^{1/2}\|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), \widehat u|_{\partial{T}}=w|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\varepsilon\|{{\mathrm{div}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}}), \widehat\sigma|_{\partial{T}}\!=\!({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\!\in\!{\mathcal{T}} \right\}. \end{align*} Moreover, for $$\widehat u\in H^{1/2}({\it{\Gamma}})$$, $$\widehat\sigma\in H^{1/2}({\it{\Gamma}})$$ we define the boundary norms \begin{align*}\|{\widehat u}\|_{1/2,{\it{\Gamma}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \varepsilon^{1/2}\|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), w|_{\it{\Gamma}}=\widehat u \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\it{\Gamma}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\varepsilon\|{{\mathrm{div}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}}), ({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{\it{\Omega}}})|_{{\it{\Gamma}}}=\widehat\sigma \right\}. \end{align*} We will need another norm in $$H^{1/2}({\it{\Gamma}})$$ defined by \begin{align*}\|{\widehat u}\|_{H^{1/2}({\it{\Gamma}})}:= \inf \left\{{ \big(\|{w}\|_{}^2 + \varepsilon \|{\nabla w}\|_{}^2\big)^{1/2}} \,:\, {w \in H^1({\it{\Omega}}), w|_{\it{\Gamma}} = \widehat u} \right\}. \end{align*} Obviously, the latter norm is weaker than the previously defined corresponding boundary norm, $$\|\,{\cdot}\,\|_{H^{1/2}({\it{\Gamma}})}\le \|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}$$. The ultra-weak formulation from Heuer & Karkulik (2017) is based on the spaces $$U$$ and $$V$$ defined by \begin{align*}\widetilde U &:= L^2({\it{\Omega}}) \times [L^2({\it{\Omega}})]^d \times L^2({\it{\Omega}}) \times H^{1/2}({\mathcal{S}}) \times H^{1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}), \\ U &:= \left\{{(u,{\boldsymbol\sigma},\rho,\widehat u^a, \widehat u^b, \widehat\sigma^a,\widehat\sigma^b) \in \widetilde U} \,:\, {\widehat u^a|_{\it{\Gamma}} = \widehat u^b|_{\it{\Gamma}}} \right\}, \\ V &:= H^1({\mathcal{T}}) \times {\boldsymbol{H}}({\mathrm{div}},{\mathcal{T}}) \times H^1({\it{\Delta}},{\mathcal{T}}), \quad\text{where}\\ H^1({\it{\Delta}},{\mathcal{T}}) &:= \{w\in H^1({\mathcal{T}}) \,:\, {\it{\Delta}} w|_T \in L^2(T) \,\forall\, T\in{\mathcal{T}}\}. \end{align*} For the analysis we need two different norms in $$U$$: \begin{align*}\|{{\boldsymbol{u}}}\|_{U,1}^2 &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \\ &\quad + \varepsilon^{3/2}\|{\widehat u^a}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon^{3/2}\|{\widehat \sigma^a}\|_{-1/2,{\mathcal{S}}}^2 + \varepsilon^{5/2}\|{\widehat \sigma^b}\|_{-1/2,{\mathcal{S}}}^2, \\ \|{{\boldsymbol{u}}}\|_{U,2}^2 &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \\ &\quad + \|{\widehat u^a}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon^{-1/2} \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 + \|{\widehat \sigma^a}\|_{-1/2,{\mathcal{S}}}^2 + \varepsilon^{1/2}\|{\widehat \sigma^b}\|_{-1/2,{\mathcal{S}}}^2 \end{align*} for $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in U$$. These norms differ in their $$\varepsilon$$-scalings of the skeleton components so that $$\|\,{\cdot}\,\|_{U,1}\le \|\,{\cdot}\,\|_{U,2}$$. In both cases, field components are measured in the so-called balanced norm$$(\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2)^{1/2}$$ which, for the exact solution, is $$(\|{u}\|_{}^2 + \varepsilon^{1/2}\|{\nabla u}\|_{}^2 + \varepsilon^{3/2} \|{{\it{\Delta}} u}\|_{}^2)^{1/2}$$ (cf. Lin & Stynes, 2012; Heuer & Karkulik, 2017). The test space $$V$$ is equipped with the norm \begin{align*}\|{{\boldsymbol{v}}}\|_{V}^2 &:= \varepsilon^{-1}\|{\mu}\|_{}^2 + \|{\nabla_{\mathcal{T}}\mu}\|_{}^2 + \varepsilon^{-1/2} \|{{\boldsymbol\tau}}\|_{}^2 + \|{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}\|_{}^2 \\ &\quad + \|{v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla_{\mathcal{T}} v}\|_{}^2 + \varepsilon^{3/2}\|{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}\|_{}^2 \quad\text{for }{\boldsymbol{v}} = (\mu,{\boldsymbol\tau},v)\in V. \end{align*} This norm is induced by the inner product $$(\cdot ,\cdot)_V$$ on $$V$$. Note that this norm is equivalent to the one defined in Heuer & Karkulik (2017). The only difference is that the term $$\varepsilon\|{\nabla_{\mathcal{T}} v}\|_{}^2$$ is not present in Heuer & Karkulik (2017). We use this norm here to get a smaller constant in Lemma 3.10 below. 3.2 Ultra-weak formulation The left- and right-hand sides of (3.1) give rise to the following definitions of the bilinear form $$b : U\times V \to {\mathbb{R}}$$ and the linear functional $$L : V \to {\mathbb{R}}$$: \begin{align*}b({\boldsymbol{u}},{\boldsymbol{v}}) &:= ({\rho} ,{\mu}) + ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}}\mu}) - \langle{\widehat \sigma^a} ,{\mu}\rangle_{\mathcal{S}} + \varepsilon^{-1/4}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u^a} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} \\ &\quad +\varepsilon^{3/4}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{3/4} \langle{\widehat\sigma^b} ,{v}\rangle_{\mathcal{S}} + ({u} ,{v}) \\ &\quad +\varepsilon^{5/4} ({\rho} ,{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) + \varepsilon^{1/4} ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{1/2} \langle{\widehat u^b} ,{\nabla_{\mathcal{T}} v\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}},\\ L({\boldsymbol{v}}) &:= ({f} ,{v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) \end{align*} for all $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b, \widehat\sigma^a,\widehat\sigma^b) \in U$$, $${\boldsymbol{v}} = (\mu,{\boldsymbol\tau},v) \in V$$. The trial-to-test operator $${\it{\Theta}}_\beta : U \to V$$ is defined as before; see (1.5). Again, the operator $$B:\;U \to V'$$ is induced by the bilinear form $$b(\cdot,\cdot)$$ and (3.1) can be written as \begin{align*}\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad B{\boldsymbol{u}} = L. \end{align*} The nontrivial kernel of $$B$$ is related to the trace operators. For the present space $$U$$ we define them by \begin{alignat*}{2}\gamma_0 &: U \to H^{1/2}({\it{\Gamma}}), &\qquad \gamma_0{\boldsymbol{u}} &:= \widehat u^a|_{\it{\Gamma}}, \\ \gamma_{\boldsymbol{n}} &: U \to H^{-1/2}({\it{\Gamma}}), &\qquad \gamma_{\boldsymbol{n}}{\boldsymbol{u}} &:= \widehat\sigma^a|_{\it{\Gamma}}. \end{alignat*} For simplicity, we consider only the symmetric formulation. The other cases can be derived similarly; see also Section 2. Analogously to the unperturbed case we introduce the nonempty convex subset \begin{align*}K^s = \left\{{ {\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{u}} \geq 0} \right\}\end{align*} and define the bilinear form $$a^s:\;U\times U\to{\mathbb{R}}$$ and linear functional $$F:\;U\to{\mathbb{R}}$$ by \begin{align*}a^s({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \tfrac12 \varepsilon^{1/4} ( \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}+\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} ), \\ F({\boldsymbol{v}}) &:= L({\it{\Theta}}_\beta{\boldsymbol{v}}) \end{align*} for $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. Here, $$\beta>0$$ is a constant to be fixed. Then, our variational inequality reads, find $${\boldsymbol{u}}\in K^s$$ such that \begin{align}\label{eq:sp:varineq}a^s({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq F({\boldsymbol{v}}-{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{v}}\in K^s. \end{align} (3.2) In the singularly perturbed case it is convenient to state coercivity and boundedness of the bilinear form $$a^s(\cdot,\cdot)$$ in the energy-based norm \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 := \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \quad ({\boldsymbol{u}}\in U). \end{align*} Note that $$\|{B\,\cdot}\|{V'}$$ is the energy norm in standard DPG settings, whereas in our case it is a seminorm. Corresponding to Theorem 2.1 we have the following result. Theorem 3.1 For all $$\beta\geq 3$$ the bilinear form $$a^s:U\times U \to {\mathbb{R}}$$ is coercive, \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 \leq C_1 a^s({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{u}}\in U, \end{align*} and bounded, \begin{align*}|a^s({\boldsymbol{u}},{\boldsymbol{v}})| \leq C_2 | | |{{\boldsymbol{u}}}| | || | |{{\boldsymbol{v}}}| | | \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}} \in U. \end{align*} The constants $$C_1,C_2>0$$ do not depend on $${\it{\Omega}}$$, $${\mathcal{T}}$$ or $$\varepsilon$$. Also $$C_1$$ is independent of $$\beta$$ but $$C_2$$ is not. Furthermore, \begin{align}\label{eq:sp:normequiv}\|{{\boldsymbol{u}}}\|_{U,1} \lesssim | | |{{\boldsymbol{u}}}| | | \lesssim \|{{\boldsymbol{u}}}\|_{U,2} \quad\text{for all } {\boldsymbol{u}}\in U, \end{align} (3.3) with generic constants that are independent of $${\mathcal{T}}$$ and $$\varepsilon$$. The variational inequality (3.2) is uniquely solvable and equivalent to problem (1.1) (setting $$c=\varepsilon$$) in the following sense. If $$u\in H^1({\it{\Omega}})$$ solves problem (1.1), then $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in K^s$$ with $${\boldsymbol\sigma} := \varepsilon^{1/4}\nabla u$$, $$\widehat u^\star|_{\partial T} := u|_{\partial T}$$, $$\widehat\sigma^\star|_{\partial T} := \sigma\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$ ($$\star\in\{a,b\}$$) solves (3.2). On the other hand, if $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in K^s$$ solves (3.2), then $$u\in H^1({\it{\Omega}})$$ solves (1.1). Moreover, the unique solution $${\boldsymbol{u}}\in K^s$$ of (3.2) satisfies \begin{align*}b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{w}}) = F({\boldsymbol{w}}) \quad\text{for all } {\boldsymbol{w}}\in U. \end{align*} We prove this result in Section 3.5. 3.3 Discretization, convergence and a posteriori error estimate We replace $$U$$ by the lowest-order subspace \begin{align*}U_h := P^0({\mathcal{T}}) \times [P^0({\mathcal{T}})]^d \times P^0({\mathcal{T}}) \times S^1({\mathcal{S}}) \times S^1({\mathcal{S}}) \times P^0({\mathcal{S}}) \times P^0({\mathcal{S}}) \end{align*} and $$K^s$$ by \begin{align*}K_h^s := \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0,\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h\geq 0} \right\}. \end{align*} The discrete version of (3.2) then reads, find $${\boldsymbol{u}}_h\in K_h^s$$ such that \begin{align}\label{eq:sp:varineqdisc}a^s({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \geq F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \quad\text{for all } {\boldsymbol{v}}_h \in K_h^s. \end{align} (3.4) With the same arguments as in Section 2.3 we can prove unique solvability. Theorem 3.2 Under the same assumptions as in Theorem 3.1, the discrete variational inequality (3.4) admits a unique solution $${\boldsymbol{u}}_h\in K_h^s$$. We have the following robust quasi-optimal a priori error estimate. Here, robustness means that the hidden constant does not depend on the positive perturbation parameter $$\varepsilon$$ (though, for simplicity, we have assumed that $$\varepsilon\le 1$$). Theorem 3.3 For $$\beta\ge 3$$ let $${\boldsymbol{u}}\in K^s$$, $${\boldsymbol{u}}_h\in K_h^s$$ denote the exact solutions of (3.2), (3.4). Then, it holds that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U,1}^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U,2}^2 + \tfrac12\varepsilon^{1/4}(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right)\!. \end{align*} The generic constant does depend on $${\it{\Omega}}$$ and $$\beta$$ but not on $${\mathcal{T}}$$ or $$\varepsilon$$. Proof. Following the proof of Theorem 2.3 we obtain (by replacing $$\|\,{\cdot}\,\|_U$$ with $$| | |\cdot| | |$$) \begin{align*}| | |{{\boldsymbol{u}}-{\boldsymbol{u}}_h}| | |^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( | | |{{\boldsymbol{u}}-{\boldsymbol{v}}_h}| | |^2 + \tfrac12\varepsilon^{1/4}(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right)\!. \end{align*} Then, (3.3) proves the error bound. □ The derivation of an a posteriori error estimate is analogous to Section 2.4. For a function $${\boldsymbol{u}}_h\in K_h^s$$ we define local error indicators by \begin{align*}\eta(T)^2 &:= \beta \|{R_T^{-1} \iota_T^*(L-B{\boldsymbol{u}}_h)}\|_{V(T)}^2, \\ \eta(E)^2 &:= \varepsilon^{1/4} \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_E, \end{align*} and the overall estimator \begin{align*}\eta^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} Here, $$V(T) := H^1(T) \times {\boldsymbol{H}}({\mathrm{div}},T) \times H^1({\it{\Delta}},T)$$ is equipped with the norm \begin{align*}\|{(\mu,{\boldsymbol\tau},v)}\|_{V(T)}^2 &:= \varepsilon^{-1}\|{\mu}\|_{T}^2 + \|{\nabla\mu}\|_{T}^2 + \varepsilon^{-1/2} \|{{\boldsymbol\tau}}\|_{T}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{T}^2 \\ &\quad{} + \|{v}\|_{T}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{T}^2 + \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{T}^2, \end{align*} where $$R_T:V(T) \to (V(T))'$$ denotes the Riesz isomorphism and $$\iota_T^*$$ is the dual of the canonical embedding $$\iota_T:V(T)\to V$$. Analogous to the proof of Theorem 2.6, in conjunction with (3.3), we obtain the following a posteriori estimate. Like the a priori estimate from Theorem 3.3, the a posteriori estimate is robust with respect to $$\varepsilon$$ ($$0<\varepsilon\le 1$$). Theorem 3.4 For $$\beta\ge 3$$ let $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ be the solutions of (3.2) and (3.4), respectively. Then, there holds the reliability estimate \begin{align*}| | |{{\boldsymbol{u}}-{\boldsymbol{u}}_h}| | | \leq C_\mathrm{rel} \eta. \end{align*} The constant $$C_\mathrm{rel}>0$$ is independent of $${\it{\Omega}}$$, $${\mathcal{T}}$$, $$\beta$$ and $$\varepsilon$$. In particular, with (3.3) we have \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U,1} \leq C_{\mathrm{rel},U} \eta, \end{align*} where $$C_{\mathrm{rel},U} := C_\mathrm{rel} C_1$$ and $$C_1$$ depends only on $${\it{\Omega}}$$. 3.4 Technical details Analogously to Lemma 2.7, we obtain boundedness of the trace operators. In this case, we use that, by definition of the norms, $$\|{\widehat u^a|_{\it{\Gamma}}}\|_{1/2,{\it{\Gamma}}}\leq \|{\widehat u^a}\|_{1/2,{\mathcal{S}}}$$ and $$\|{\widehat\sigma^a|_{\it{\Gamma}}}\|_{-1/2,{\it{\Gamma}}}\leq \|{\widehat\sigma^a}\|_{-1/2,{\mathcal{S}}}$$. Lemma 3.5 The operators $$\gamma_0 : \big(U,\|\,{\cdot}\,\|_{U,2}\big) \to \big(H^{1/2}({\it{\Gamma}}),\|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}\big)$$ and $$\gamma_{\boldsymbol{n}} : \big(U,\|\,{\cdot}\,\|_{U,2}\big) \to \big( H^{-1/2}({\it{\Gamma}}), \|\,{\cdot}\,\|_{-1/2,{\it{\Gamma}}}\big)$$ have unit norm. We now adapt the definition of the previously employed extension operator $$\mathcal{E}$$ to the current situation. For a function $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ we define its quasi-harmonic extension $$\widetilde u \in H^1({\it{\Omega}})$$ as the unique solution of \begin{align}\label{eq:sp:harmext}-\varepsilon {\it{\Delta}} \widetilde u + \widetilde u &=0 \quad\text{in }{\it{\Omega}}, \qquad \widetilde u|_{\it{\Gamma}} = \widehat v, \end{align} (3.5) and define $$\mathcal{E} : H^{1/2}({\it{\Gamma}}) \to U$$ by \begin{align*}\mathcal{E} \widehat v := \left(\widetilde u, \varepsilon^{1/4}\nabla \widetilde u, \varepsilon^{1/4} {\it{\Delta}} \widetilde u, \widetilde u|_{{\mathcal{S}}}, \widetilde u|_{{\mathcal{S}}}, \varepsilon^{1/4} \left(\nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T}\right)_{T\in{\mathcal{T}}}, \varepsilon^{1/4} \left(\nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T}\right)_{T\in{\mathcal{T}}}\right). \end{align*} This operator characterizes the kernel of $$B$$. Note that, in contrast to $$\|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}$$, the norm $$\|\,{\cdot}\,\|_{H^{1/2}({\it{\Gamma}})}$$ (recall the definitions in Section 3.1) is inherited from the energy norm associated with problem (3.5). Lemma 3.6 For given $$\widehat v\in H^{1/2}({\it{\Gamma}}),$$ let $$\widetilde u \in H^1({\it{\Omega}})$$ be the unique solution of (3.5). Then, \begin{align*}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2 = \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 = \varepsilon \langle{\partial_{\boldsymbol{n}} \widetilde u} ,{\widehat v}\rangle_{\it{\Gamma}} = \varepsilon^{3/4}\langle{\gamma_{\boldsymbol{n}} \mathcal{E} \widehat v} ,{\widehat v}\rangle_{\it{\Gamma}}. \end{align*} Proof. The last identity follows by definition of the operator $$\mathcal{E}$$. Using the weak formulation of problem (3.5) we have \begin{align*}\varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 &= \varepsilon\langle{\partial_{\boldsymbol{n}}\widetilde u} ,{\widehat v}\rangle_{\it{\Gamma}}= \varepsilon({\nabla \widetilde u} ,{\nabla w}) + ({\widetilde u} ,{w}) \\ &\leq \Big( \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2\Big)^{1/2}\Big( \varepsilon\|{\nabla w}\|_{}^2 + \|{w}\|_{}^2\Big)^{1/2}\end{align*} for all $$w\in H^1({\it{\Omega}})$$ with $$w|_{\it{\Gamma}} = \widehat v$$. Thus, $$\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2 = \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2$$. □ Similarly to Lemma 2.9, there holds the following result in the singularly perturbed case. Lemma 3.7 The operators $$B : U \to V'$$, $$\mathcal{E}: H^{1/2}({\it{\Gamma}})\to U$$ have the following properties: (i) The operators $$B$$ and $$\mathcal{E}$$ are bounded: \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'} &\lesssim \|{{\boldsymbol{u}}}\|_{U,2} \quad\text{for all }{\boldsymbol{u}}\in U, \qquad \|{\mathcal{E} \widehat v}\|_{U,1} \simeq \varepsilon^{-1/4} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}\quad\text{for all } \widehat v \in H^{1/2}({\it{\Gamma}}). \end{align*} The generic constants are independent of $${\mathcal{T}}$$ and $$\varepsilon$$. (ii) The kernel of $$B$$ consists of all quasi-harmonic extensions, i.e., $$\ker(B) = \operatorname{ran}(\mathcal{E})$$. (iii) The operator $$\mathcal{E}$$ is a right inverse of $$\gamma_0$$. (iv) The operator $$B:\;U/\ker(B)\to V'$$ is inf–sup stable: \begin{align*}\|{{\boldsymbol{\it{u}}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1} \lesssim \|{B{\boldsymbol{u}}}\|_{V'} = \sup_{{\boldsymbol{\it{v}}}\in V} \frac{\langle B\mathbf{u},{\boldsymbol{v}}\rangle }{\|{\boldsymbol{v}}{{\|}_{V}}}= b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align*} The generic constant depends on $${\it{\Omega}}$$ but not on $${\mathcal{T}}$$ or $$\varepsilon$$. Proof. First, by Heuer & Karkulik (2017, Lemma 3), we have $$b({\boldsymbol{u}},{\boldsymbol{v}})\lesssim\|{{\boldsymbol{u}}}\|_{U,2}\|{{\boldsymbol{v}}}\|_{V}$$ for all $${\boldsymbol{u}}\in U,{\boldsymbol{v}}\in V$$. Dividing by $$\|{{\boldsymbol{v}}}\|_{V}$$ and taking the supremum proves boundedness of $$B$$. Second, for $$\widehat v\in H^{1/2}({\it{\Gamma}})$$, let $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b) = \mathcal{E} \widehat v$$. By definition of the skeleton norms we have \begin{align*}&\|{\widehat u^\star}\|_{1/2,{\mathcal{S}}}^2 \leq \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2,\quad\|{\widehat \sigma^\star}\|_{-1/2,{\mathcal{S}}}^2 \leq \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon\|{\rho}\|_{}^2 \quad (\star\in\{a,b\}). \end{align*} In the definition of $$\|\,{\cdot}\,\|_{U,1}$$, these norms are scaled with positive powers of $$\varepsilon$$. Thus, \begin{align*}\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \leq \|{{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2. \end{align*} Using that $$\rho = \varepsilon^{1/4} {\it{\Delta}} u = \varepsilon^{1/4} \varepsilon^{-1} u = \varepsilon^{-3/4} u$$ and $${\boldsymbol\sigma} = \varepsilon^{1/4}\nabla u$$ by the definition of $${\boldsymbol{u}}=\mathcal{E}\widehat v$$ (cf. (3.5)) we obtain \begin{align*}\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon\|{\rho}\|_{}^2 = \|{u}\|_{}^2 + \varepsilon^{1/2} \|{\nabla u}\|_{}^2 + \varepsilon^{-1/2}\|{u}\|_{}^2 \simeq \varepsilon^{-1/2}\left( \|{u}\|_{}^2 + \varepsilon \|{\nabla u}\|_{}^2 \right). \end{align*} The last term on the right-hand side is equal to $$\varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2$$ by Lemma 3.6. Finally, (ii), (iii) and (iv) are proved in Führer & Heuer (2016, Lemmas 3, 4). □ Similarly to Lemma 2.11 (for the unperturbed case), we need to control the Neumann traces of elements of the quotient space $$U/\ker(B)$$. As we have seen, this has a fundamental relation to the stability of the homogeneous adjoint problem with prescribed Dirichlet boundary condition; cf. Lemma 2.10. In the singularly perturbed case, the situation is a little more technical. Following Heuer & Karkulik (2017) (see Lemmas 8 and 9 there), we split the stability analysis of the adjoint problem into two parts. These are the following Lemmas 3.8 and 3.9. The last lemma of this section (Lemma 3.10) then states the control of the Neumann traces. Lemma 3.8 Let $$\widehat w\in H^{1/2}({\it{\Gamma}})$$ be given. The problem \begin{align}{\mathrm{div}}{\boldsymbol\lambda} + \varepsilon^{-1/2}w &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualestimate:a} \\ \end{align} (3.6a) \begin{align}{\boldsymbol\lambda} + \nabla w &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualestimate:b} \\ \end{align} (3.6b) \begin{align}w|_{{\it{\Gamma}}} &= \widehat w \label{eq:sp:dualestimate:c}\end{align} (3.6c) admits a unique solution $$(w,{\boldsymbol\lambda}) \in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ with $${\it{\Delta}} w\in H^1({\it{\Omega}})$$, and \begin{align*}\|{{\boldsymbol\lambda}}\|_{}^2 + \varepsilon^{1/2}\|{{\mathrm{div}}{\boldsymbol\lambda}}\|_{}^2 = \|{\nabla w}\|_{}^2 + \varepsilon^{-1/2}\|{w}\|_{}^2 \leq \varepsilon^{-1} \|{\widehat w}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Proof. The proof follows the same arguments as given in the proof of Lemma 2.10. To see the estimate for the norms, we make use of the weak formulation and obtain \begin{align*}\|{\nabla w}\|_{}^2 + \varepsilon^{-1/2}\|{w}\|_{}^2 \leq \|{\nabla \widetilde u}\|_{}^2 + \varepsilon^{-1/2}\|{\widetilde u}\|_{}^2 \leq \varepsilon^{-1} \left( \varepsilon\|{\nabla\widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 \right) \end{align*} for all $$\widetilde u\in H^1({\it{\Omega}})$$ with $$\widetilde u|_{\it{\Gamma}} = \widehat w$$. Taking the infimum over these functions $$\widetilde u$$ finishes the proof. □ Lemma 3.9 Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ be given. The problem \begin{align}{\mathrm{div}}{\boldsymbol\tau} + v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:a} \\ \end{align} (3.7a) \begin{align}\nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau} &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:b}\\ \end{align} (3.7b) \begin{align}\varepsilon^{5/4} {\it{\Delta}} v + \mu &= 0\quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:c} \\ \end{align} (3.7c) \begin{align}\quad v|_{\it{\Gamma}} = 0, \qquad {\it{\Delta}} v|_{\it{\Gamma}} = -\varepsilon^{-5/4} {\widehat v}\end{align} (3.7d) has a unique solution $${\boldsymbol{v}}:=(\mu,{\boldsymbol\tau},v)\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})\times H^1({\it{\Delta}},{\it{\Omega}})$$. It satisfies $$\mu|_{\it{\Gamma}} = \widehat v$$ and \begin{align*}\|{{\boldsymbol{v}}_{V}}\| \leq 3/\sqrt{2} \varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} Proof. Let $$(w,{\boldsymbol\lambda})\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ be the solution of (3.6) with $$\widehat w = \varepsilon^{-1/4}\widehat v$$. Define $$v\in H_0^1({\it{\Omega}})$$ to be the unique solution of \begin{align*}-\varepsilon {\it{\Delta}} v + v = w\quad\text{in } {\it{\Omega}}, \qquad v|_{\it{\Gamma}} = 0. \end{align*} Note that $${\it{\Delta}} v = \varepsilon^{-1} (v-w) \in H^1({\it{\Omega}})$$ and $${\it{\Delta}} v|_{\it{\Gamma}} = -\varepsilon^{-1}w|_{\it{\Gamma}} = -\varepsilon^{-5/4}\widehat v$$. We define $$\mu:= -\varepsilon^{5/4}{\it{\Delta}} v\in H^1({\it{\Omega}})$$ and have $$\mu|_{\it{\Gamma}} = \varepsilon^{1/4}w|_{\it{\Gamma}} = \widehat v$$. In particular, (3.7c) and (3.7d) are satisfied. Now, define $${\boldsymbol\tau} := \varepsilon^{1/2}{\boldsymbol\lambda} - \varepsilon\nabla v$$. Note that $${\boldsymbol\lambda}\in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ and $$\nabla v \in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$. Hence, $${\boldsymbol\tau}\in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$. Together with (3.6a) and $$-\varepsilon{\it{\Delta}} v + v = w$$ we get \begin{align*}{\mathrm{div}}{\boldsymbol\tau} = \varepsilon^{1/2}{\mathrm{div}}{\boldsymbol\lambda} - \varepsilon{\it{\Delta}} v = -w - \varepsilon{\it{\Delta}} v = -v, \end{align*} which is (3.7a). One also establishes that (3.7b) holds. In fact, by the definition of $${\boldsymbol\tau}$$ and relation (3.6b), we find \begin{align*}\nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau}&= \nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4} (\varepsilon^{1/2}{\boldsymbol\lambda}-\varepsilon\nabla v)\\ &= \nabla\mu + \varepsilon^{1/4}\nabla v - \varepsilon^{1/4}\nabla w. \end{align*} The last term vanishes since $$\mu=-\varepsilon^{5/4}{\it{\Delta}} v = -\varepsilon^{1/4}(v-w)$$ by definition of $$\mu$$ and $$v$$. Now, testing (3.7c) with $$\varepsilon^{1/4}{\it{\Delta}} z$$, (3.7b) with $$\varepsilon^{1/4}\nabla z$$ and (3.7a) with $$z$$ for $$z\in H^1({\it{\Delta}},{\it{\Omega}})$$ with $$z|_{\it{\Gamma}} = 0$$, and adding the resulting equations, we obtain, after integrating by parts, \begin{align}\label{eq:sp:dualproblem:varform}\varepsilon^{3/2}({{\it{\Delta}} v} ,{{\it{\Delta}} z}) + (\varepsilon^{1/2}+\varepsilon) ({\nabla v} ,{\nabla z}) + ({v} ,{z}) = -\varepsilon^{1/4} \langle{\nabla z\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{\widehat v}\rangle_{\it{\Gamma}}. \end{align} (3.8) For any $$\widetilde u\in H^1({\it{\Omega}})$$ with $$\widetilde u|_{\it{\Gamma}} = \widehat v$$ we infer \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 &= -\varepsilon^{1/4}\langle{\nabla v\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{\widehat v}\rangle_{\it{\Gamma}}= -\varepsilon^{1/4} ({{\it{\Delta}} v} ,{\widetilde u}) -\varepsilon^{1/4}({\nabla v} ,{\nabla\widetilde u)} \\ &\leq \varepsilon^{3/4}\|{{\it{\Delta}} v}\|_{} \|{\varepsilon^{-1/2}\widetilde u}\|_{} + \varepsilon^{1/4}\|{\nabla v}\|_{}\|{\nabla \widetilde u}\|_{} \\ &\leq \left(\varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \varepsilon^{1/2}\|{\nabla v}\|_{}^2\right)^{1/2}\varepsilon^{-1/2}\left( \|{\widetilde u}\|_{}^2 + \varepsilon \|{\nabla \widetilde u}\|_{}^2\right)^{1/2}. \end{align*} On the one hand, we conclude \begin{align}\label{eq:Hm12estimate:estV1}\left(\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right)^{1/2}\leq \varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}. \end{align} (3.9) On the other hand, using Young’s inequality, we also conclude that \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 \leq \tfrac{\delta^{-1}}2 \left(\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + \varepsilon^{1/2}\|{\nabla v}\|_{}^2\right) + \tfrac{\delta}2 \varepsilon^{-1} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} For $$\delta=\tfrac12$$ we get \begin{align}\label{eq:Hm12estimate:estV2}\varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 \leq \tfrac14 \varepsilon^{-1}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align} (3.10) By (3.7c) and (3.7a) we have \begin{align*}\varepsilon^{-1}\|{\mu}\|_{}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{}^2 = \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \|{v}\|_{}^2. \end{align*} It remains to estimate the norms of $${\boldsymbol\tau}$$ and $$\nabla v$$. To this end, we rewrite the term $$({{\boldsymbol\lambda}} ,{\nabla v})$$. Integrating by parts, the condition $$v|_{\it{\Gamma}} = 0$$, (3.6a) and the identity $$w=-\varepsilon{\it{\Delta}} v + v$$ show that \begin{align*}({{\boldsymbol\lambda}} ,{\nabla v}) = -({{\mathrm{div}}{\boldsymbol\lambda}} ,{v}) &= \varepsilon^{-1/2}({w} ,{v}) = -\varepsilon^{1/2}({{\it{\Delta}} v} ,{v}) + \varepsilon^{-1/2}({v} ,{v}) = \varepsilon^{-1/2}\left( \varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right). \end{align*} Recall that $$\varepsilon^{-1/4}{\boldsymbol\tau} = \varepsilon^{1/4}{\boldsymbol\lambda} -\varepsilon^{3/4}\nabla v$$. Thus, \begin{align*}\varepsilon^{-1/2}\|{{\boldsymbol\tau}}\|_{}^2 = \varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2 -2\varepsilon({{\boldsymbol\lambda}} ,{\nabla v}) + \varepsilon^{3/2}\|{\nabla v}\|_{}^2. \end{align*} For the estimation of $$\|{\nabla \mu}\|_{}$$ we use (3.7b) and again $$\varepsilon^{1/4}{\boldsymbol\lambda} = \varepsilon^{3/4}\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau}$$ to obtain \begin{align*}\|{\nabla\mu}\|_{}^2 = \|{\varepsilon^{1/4}\nabla v + \varepsilon^{1/4}{\boldsymbol\lambda}}\|_{}^2 = \varepsilon^{1/2}\|{\nabla v}\|_{}^2 + 2\varepsilon^{1/2}({{\boldsymbol\lambda}} ,{\nabla v}) + \varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2. \end{align*} This gives the estimate \begin{align*}&\varepsilon^{-1}\|{\mu}\|_{}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{}^2 + \varepsilon^{-1/2}\|{{\boldsymbol\tau}}\|_{}^2 + \|{\nabla\mu}\|_{}^2 \\ &\qquad \leq \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \|{v}\|_{}^2 + 2\varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2 + (\varepsilon^{1/2}+2\varepsilon)\|{\nabla v}\|_{}^2 + 2\|{v}\|_{}^2 \\ &\qquad \leq \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 + 2\left(\varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right) + 2\varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2. \end{align*} Using estimates (3.9), (3.10) and Lemma 3.8, we put everything together to conclude for the overall norm \begin{align*}\|{(\mu,{\boldsymbol\tau},v)}\|_{V}^2 \leq \frac{9}2 \varepsilon^{-1}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} To see uniqueness of $${\boldsymbol{v}}$$, let $$(\mu_2,{\boldsymbol\tau}_2,v_2)$$ solve (3.7). Define $${\boldsymbol{w}}:={\boldsymbol{v}}-(\mu_2,{\boldsymbol\tau}_2,v_2)$$ and set $$w:=v-v_2$$. Note that $$w|_{\it{\Gamma}} = 0$$ and $${\it{\Delta}} w|_{\it{\Gamma}} = 0$$. The variational formulation for $$w$$, which can be obtained in the same way as (3.8), proves \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} w}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla w}\|_{}^2 + \|{w}\|_{}^2 = 0. \end{align*} Thus, $$w=0$$ or equivalently $$v=v_2$$. Equation (3.7c) then gives $$\mu = -\varepsilon^{5/4}{\it{\Delta}} v = -\varepsilon^{5/4}{\it{\Delta}} v_2 = \mu_2$$ and, similarly, (3.7b) shows $${\boldsymbol\tau}={\boldsymbol\tau}_2$$. □ Lemma 3.10 It holds that \begin{align*}\varepsilon^{1/4}|\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\widehat v}\rangle_{\it{\Gamma}}| \leq 3/\sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'}\varepsilon^{-1/4}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})} \quad\text{for all } {\boldsymbol{u}}\in U, \, \widehat v\in H^{1/2}({\it{\Gamma}}). \end{align*} Proof. The idea of the proof is the same as in the proof of Lemma 2.11, using Lemma 3.9 instead of Lemma 2.10. □ 3.5 Proof of Theorem 3.1 First, we show boundedness and coercivity of $$a^s(\cdot,\cdot)$$ with respect to the norm $$| | |\cdot| | |$$. Then, the Lions–Stampacchia theorem (see, e.g., Glowinski et al., 1981; Rodrigues, 1987; Glowinski, 2008) proves unique solvability of (2.4), provided that $$F : U\to {\mathbb{R}}$$ is a linear functional, which follows by the boundedness of $${\it{\Theta}}:U\to{\mathbb{R}}$$. We start by showing the boundedness. Since $$b(\cdot,{\it{\Theta}}\cdot)$$ is symmetric and positive semidefinite, the Cauchy–Schwarz inequality proves \begin{align*}|b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}})| \leq \beta b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} b({\boldsymbol{v}},{\it{\Theta}}{\boldsymbol{v}})^{1/2} = \beta \|{B{\boldsymbol{u}}}\|_{V'}\|{B{\boldsymbol{v}}}\|_{V'}\end{align*} for all $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. For the boundary terms, we consider \begin{align*}\varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq \varepsilon^{1/4}|\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| + \varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}|. \end{align*} The first term on the right-hand side is estimated with Lemma 3.10. Let $$\widetilde u \in H^1({\it{\Omega}})$$ be the quasi-harmonic extension of $$\gamma_0{\boldsymbol{u}}$$ and let $$\widetilde v \in H^1({\it{\Omega}})$$ be the quasi-harmonic extension of $$\gamma_0{\boldsymbol{v}}$$. Then, for the second term, we get with integration by parts (cf. Lemma 3.6), \begin{align*}\varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}} = \varepsilon^{-1/2}\Big( \varepsilon({\nabla \widetilde u} ,{\nabla\widetilde v}) + ({\widetilde u} ,{\widetilde v}) \Big) \leq \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} Together, this gives for the boundary term, \begin{align*}\varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq 3/\sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'}\varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})} + \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} The second boundary term $$\varepsilon^{1/4} \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}$$ is treated identically. Altogether this proves the boundedness of $$a^s(\cdot,\cdot)$$. For the proof of coercivity we use Lemma 3.6, Lemma 3.10 and Young’s inequality to find that, for $$\delta>0$$, \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 &= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 = \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}(\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &\leq \|{B{\boldsymbol{u}}}\|_{V'}^2 + 3/\sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &\leq (1+\delta^{-1}\tfrac{9}4) \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}+\tfrac\delta2 \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} We choose $$\delta = \tfrac{9}8$$, which implies $$1+\delta^{-1}\tfrac{9}4 = 3$$. Then, subtracting the last term on the right-hand side, we obtain for $$\beta\ge 3$$, \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \tfrac{7}{16} \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \leq 3 \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\leq a^s({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all }{\boldsymbol{u}}\in U. \end{align*} Next we show (3.3). By Lemma 3.7 and the triangle inequality we obtain \begin{align*}\|{{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1}^2 + \|{\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} For the proof of the upper bound in (3.3) let $$\widetilde u\in H^1({\it{\Omega}})$$ be a function that attains the minimum in the definition of $$\|{\widehat u^b}\|_{1/2,{\mathcal{S}}}$$. Then, \begin{align*}\varepsilon^{-1/2} \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 = \varepsilon^{-1/2}\big( \|{\widetilde u}\|_{}^2 + \varepsilon^{1/2}\|{\nabla \widetilde u}\|_{}^2 \big) \geq \varepsilon^{-1/2} \big( \|{\widetilde u}\|_{}^2 + \varepsilon\|{\nabla \widetilde u}\|_{}^2 \big). \end{align*} Since $$\widetilde u|_{\it{\Gamma}} = \widehat u^b|_{\it{\Gamma}} = \widehat u^a|_{\it{\Gamma}} = \gamma_0{\boldsymbol{u}}$$, the right-hand side is an upper bound of $$\varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2$$. Thus, together with Lemma 3.7, we get \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 = \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \lesssim \|{{\boldsymbol{u}}}\|_{U,2}^2 + \varepsilon^{-1/2}\|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 \lesssim \|{{\boldsymbol{u}}}\|_{U,2}^2. \end{align*} The remainder of the proof follows the same arguments as in the proof of Theorem 2.1. 4 Examples In this section, we present various numerical examples in two dimensions ($$d=2$$). For the first example in Section 4.2 we take a manufactured solution $$u\in H^2({\it{\Omega}})$$. The standard finite element method with lowest-order discretization on quasi-uniform meshes converges at a rate $${\mathcal{O}}(h)$$, where $$h$$ denotes the diameter of elements in $${\mathcal{T}}$$. We observe the same optimal rate for the DPG methods with $$\star\in\{0,{\boldsymbol{n}},s\}$$, analysed in Section 2. In Section 4.3, we consider an L-shaped domain with unknown solution and an expected singularity at the reentrant corner. Indeed, we will see that a uniform method gives suboptimal convergence, whereas an adaptive method driven by the estimator $$\eta$$ from Section 2.4 recovers the optimal one. Finally, in Section 4.4, we use a family of manufactured solutions that exhibit typical boundary layers for the reaction-dominated diffusion problem. Our numerical results underline the robustness of the a posteriori error estimate, as stated by Theorem 3.4. 4.1 General setting As is usual for DPG methods we replace the infinite-dimensional test space $$V$$ used in the calculation of optimal test functions (1.5) by a finite-dimensional subspace $$V_h$$; that is, we replace the test function $${\it{\Theta}}_\beta{\boldsymbol{u}}_h$$ for $${\boldsymbol{u}}_h\in U_h$$ by $${\it{\Theta}}_{\beta,h}{\boldsymbol{u}}_h$$ defined through \begin{align*}\left( {{{\it{\Theta}}_{\beta,h}{\boldsymbol{u}}_h},{{\boldsymbol{v}}_h}} \right)_V = \beta b({\boldsymbol{u}}_h,{\boldsymbol{v}}_h) \quad\text{for all }{\boldsymbol{v}}_h\in V_h. \end{align*} Here we choose \begin{align*}V_h := \begin{cases}P^2({\mathcal{T}})\times[P^2({\mathcal{T}})]^2 & \text{for the methods from Section 2}, \\ P^2({\mathcal{T}})\times[P^2({\mathcal{T}})]^2 \times P^4({\mathcal{T}}) & \text{for the method from Section 3}. \end{cases}\end{align*} These choices are motivated by Gopalakrishnan & Qiu (2014). We refer the interested reader to this work for more details. The resulting DPG scheme is called the practical DPG method. For the scaling parameter of the test functions, we choose $$\beta=2$$ for the methods from Section 2 and $$\beta=3$$ for the method from Section 3. We use the standard basis for the lowest-order spaces $$U_h$$, i.e., the element characteristic functions for $$P^0({\mathcal{T}})$$, $$P^0({\mathcal{S}})$$, and nodal basis functions (hat functions) for $$S^1({\mathcal{S}})$$. These choices allow a simple implementation of the inequality constraints in the cones $$K_h^\star$$. We solve the discrete variational inequalities (2.6), (3.4) with a (primal–dual) active set algorithm (see Hoppe & Kornhuber, 1994; Kärkkäinen et al., 2003). More precisely, we implemented a modification of Kärkkäinen et al. (2003, Algorithm A1). (which deals with obstacle problems) for the present problem (here, we consider inequality constraints only for degrees of freedom that are associated with the boundary). For the problems where the solution is known in analytical form, we compute different error quantities depending on the underlying problem from Section 2 or 3. Section 2: Let $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)$$ denote the exact solution of (2.4) and let $${\boldsymbol{u}}_h=(u_h,{\boldsymbol\sigma}_h,\widehat u_h,\widehat\sigma_h)$$ be its approximation. We define \begin{alignat*}{2}\quad\qquad\text{err}(u) &:= {\|u-u_{h}\|}{}, &\quad \text{err}({\boldsymbol\sigma}) &:= \left\| {{{\boldsymbol\sigma}-{\boldsymbol\sigma}_h}} \right\|{}, \\ \quad\qquad\text{err}(\widehat u) &:= \left(\left\| {{u-\widetilde u_h}} \right\|{}^2 + \left\| {{\nabla(u-\widetilde u_h)}} \right\|{}^2\right)^{1/2}, &\quad \text{err}(\widehat\sigma) &:= \left(\left\| {{{\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h}} \right\|{}^2 + \left\| {{{{\rm div\,}}({\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h)}} \right\|{}^2\right)^{1/2}. \end{alignat*} Here, $$\widetilde u_h\in S^1({\mathcal{T}})$$ is the nodal interpolant of $$\widehat u_h$$ at the nodes of $${\mathcal{T}}$$. Similarly, $$\widetilde{\boldsymbol\sigma}_h$$ is the Raviart–Thomas interpolation of $$\widehat \sigma_h$$. Then, it follows by the definition of the trace norms, \begin{align*}\left\| {{{\boldsymbol{u}}-{\boldsymbol{u}}_h}} \right\|_{U} \leq \left(\text{err}(u)^2 + \text{err}({\boldsymbol\sigma})^2 + \text{err}(\widehat u)^2 + \text{err}(\widehat\sigma)^2\right)^{1/2}. \end{align*} Section 3: Let $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)$$ denote the exact solution of (3.2) and let $${\boldsymbol{u}}_h=(u_h,{\boldsymbol\sigma}_h,\rho_h,\widehat u_h^a,\widehat u_h^b,\widehat\sigma_h^a,\widehat\sigma_h^b)$$ be its approximation. Define $$\text{err}(u)$$ and $$\text{err}({\boldsymbol\sigma})$$ as above and additionally \begin{align*}\text{err}(\widehat u^\star) &:= \left(\left\| {{u-\widetilde u_h^\star}} \right\|{}^2 + \varepsilon^{1/2}\left\| {{\nabla(u-\widetilde u_h^\star)}} \right\|{}^2\right)^{1/2}, \\ \text{err}(\widehat\sigma^\star) &:= \left(\left\| {{{\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h^\star}} \right\|{}^2 + \varepsilon\left\| {{{{\rm div\,}}({\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h^\star)}} \right\|{}^2\right)^{1/2}, \\ \text{err}(\rho) &:= \varepsilon^{1/2}\left\| {{\rho-\rho_h}} \right\|{}, \end{align*} for $$\star\in\{a,b\}$$, and $$\widetilde u_h^\star$$, $$\widetilde\sigma_h^\star$$ defined in the same way as above. Our total error estimator is \begin{align*}\text{err}({\boldsymbol{u}}) &:= \Big( \text{err}(u)^2 + \text{err}({\boldsymbol\sigma})^2 + \text{err}(\rho)^2 \\ &\quad +\varepsilon^{3/2} \text{err}(\widehat u^a)^2 + \varepsilon \text{err}(\widehat u^b)^2 + \varepsilon^{3/2}\text{err}(\widehat\sigma^a)^2+\varepsilon^{5/2}\text{err}(\widehat\sigma^b)^2 \Big)^{1/2}\end{align*} so that \begin{align*}\left\| {{{\boldsymbol{u}}-{\boldsymbol{u}}_h}} \right\|_{U,1} \leq \text{err}({\boldsymbol{u}}). \end{align*} For examples with singularities and/or boundary or interior layers, we use a standard adaptive algorithm that uses $$\eta(T)$$ and $$\eta(E)$$ to mark elements by the bulk criterion. For convenience, we define $$\eta({\mathcal{T}})$$ and $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ by \begin{align*}\eta({\mathcal{T}})^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2, \quad\eta({\mathcal{S}}_{\it{\Gamma}})^2 := \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} 4.2 Piecewise smooth solution (Section 2) We consider the domain $${\it{\Omega}}:=(-1,1)\times (0,1)$$ and the manufactured solution \begin{align*}u(x,y) := \begin{cases}-16x^2(1-x)y(1-y), & x\geq 0, \\ 2(x+1)^3 - 3(x+1)^2 +1, & x<0. \end{cases}\end{align*} This solution satisfies $$u(x,y) = 0$$ on the part of $${\it{\Gamma}}=\partial{\it{\Omega}}$$ where $$x\geq 0$$ and $$\partial_{{\boldsymbol{n}}_{\it{\Omega}}}u = 0$$ on the part of $${\it{\Gamma}}$$ where $$x<0$$. We calculate $$f:=-{\it{\Delta}} u + u$$ and note that $$u\in H^1({\it{\Delta}},{\it{\Omega}})$$. Also note that $$u(x,y)$$ is smooth in both regions $$x>0$$ and $$x<0$$. Moreover, observe that $$u$$ satisfies the Signorini problem (1.1) with $$c=1$$. Our initial mesh consists of $$8$$ congruent triangles. We solve (2.6) for $$\star \in \{0,{\boldsymbol{n}},s\}$$ and plot the errors $$\text{err}(u)$$, $$\text{err}({\boldsymbol\sigma})$$, $$\text{err}(\widehat u)$$ and $$\text{err}(\widehat\sigma)$$ for a sequence of uniformly refined triangulations. Moreover, in the case $$\star=s$$, we compare these error quantities with the reliable error estimator $$\eta$$. The results are given in Fig. 1 for $$\star=0$$, $$\star={\boldsymbol{n}}$$ and Fig. 2 for $$\star=s$$. We observe optimal convergence rates $${\mathcal{O}}(h^\alpha) = {\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ with $$\alpha=1$$ for the error quantities. This rate is visualized by a triangle. In Fig. 2, we see that also the estimator $$\eta({\mathcal{T}})$$ converges with this rate whereas $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ has a higher convergence rate of approximately $$\alpha = 2.8$$. Fig. 1. View largeDownload slide Error quantities for the example from Section 4.2 with $$\star=0$$ (left) and $$\star={\boldsymbol{n}}$$ (right). Fig. 1. View largeDownload slide Error quantities for the example from Section 4.2 with $$\star=0$$ (left) and $$\star={\boldsymbol{n}}$$ (right). Fig. 2. View largeDownload slide Error quantities and estimators for the example from Section 4.2 with $$\star=s$$. Fig. 2. View largeDownload slide Error quantities and estimators for the example from Section 4.2 with $$\star=s$$. 4.3 Unknown solution (Section 2) Let $${\it{\Omega}} = (-1,1)^2\setminus [-1,0]^2$$ with initial triangulation visualized in Fig. 3. We define Fig. 3. View largeDownload slide L-shaped domain with initial triangulation of $$12$$ elements. Fig. 3. View largeDownload slide L-shaped domain with initial triangulation of $$12$$ elements. \begin{align*}f(x,y) := \begin{cases}-1, & |(x,y)|\leq 0.8, \\ \tfrac12 & \text{otherwise}. \end{cases}\end{align*} For this right-hand side, the solution is not known to us in analytical form. Therefore, we compute only the error estimators. The results are plotted in Fig. 4. We observe that uniform refinement leads to a reduced order of convergence $${\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ of approximately $$\alpha=0.7$$, whereas adaptive refinement regains the optimal order $$\alpha=1$$. This is a strong indicator that the unknown solution has a singularity at the reentrant corner, which is what one expects. Figure 5 visualizes meshes at different steps of the adaptive loop and supports this observation. Fig. 4. View largeDownload slide Estimators for the example from Section 4.3 for uniform and adaptive refinement. Fig. 4. View largeDownload slide Estimators for the example from Section 4.3 for uniform and adaptive refinement. Fig. 5. View largeDownload slide Meshes at different steps in the adaptive algorithm for the problem from Section 4.3. Fig. 5. View largeDownload slide Meshes at different steps in the adaptive algorithm for the problem from Section 4.3. 4.4 Piecewise smooth solution with boundary layer (Section 3) Let $${\it{\Omega}}:=(-1,1)\times (0,1)$$ with manufactured solution \begin{align*}u(x,y) := \begin{cases}-x^2\left(e^{-2(1-x)/\sqrt\varepsilon}\right) \left( e^{-y/\sqrt\varepsilon} + e^{-(1-y)/\sqrt\varepsilon} - e^{-1/\sqrt\varepsilon} - 1\right), &x\geq 0, \\ 2(x+1)^3 - 3(x+1)^2 +1, & x<0. \end{cases}\end{align*} We choose $$f=-\varepsilon{\it{\Delta}} u+u$$. Observe that $$u$$ satisfies (1.1) with $$c=\varepsilon$$. In particular, $$u$$ has a layer of order $$\sqrt{\varepsilon}$$ at the boundary for $$x\geq 0$$. We solve the variational inequality (3.4) on a sequence of adaptively refined triangulations for $$\varepsilon\in\{10^{-2},10^{-4},10^{-6},10^{-8}\}$$. We start with a coarse initial triangulation that consists of only $$\#{\mathcal{T}}_0 = 8$$ congruent triangles. In Fig. 6, we compare the estimator $$\eta$$ with the total error $$\text{err}({\boldsymbol{u}})$$. As in the works Heuer & Karkulik (2017) and Führer & Heuer (2016), we observe that, after boundary layers have been resolved, our method leads to an optimal convergence rate $${\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ ($$\alpha=1$$). In the pre-asymptotic range, we see that there is a small gap between the curves representing $$\eta({\mathcal{T}})$$ and $$\text{err}({\boldsymbol{u}})$$. This is due to the fact that the optimal test functions cannot be approximated accurately enough in the test space $$V_h$$. Similar observations have been made in Heuer & Karkulik (2017) and Führer & Heuer (2016). After layers are resolved we see that the curves are close together and uniformly in $$\varepsilon$$ (for the selected values). This confirms the robustness of the a posteriori estimate by Theorem 3.4. We also see that the boundary estimator $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ is small in comparison with $$\eta({\mathcal{T}})$$, and has a higher convergence rate. Fig. 6. View largeDownload slide Comparison of total error $$\text{err}({\boldsymbol{u}})$$ and estimators $$\eta({\mathcal{T}})$$, $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ for the problem from Section 4.4. Fig. 6. View largeDownload slide Comparison of total error $$\text{err}({\boldsymbol{u}})$$ and estimators $$\eta({\mathcal{T}})$$, $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ for the problem from Section 4.4. Our primary interest was to construct a robust method in the sense that the error in the balanced norm of the field variables $$u,{\boldsymbol\sigma},\rho$$ is controlled uniformly in $$\varepsilon$$. The error estimator $$\eta$$ does precisely this, as stated by Theorem 3.4 and seen in Fig. 6. To further underline this statement, Fig. 7 shows the ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$. We observe that, again after boundary layers have been resolved, this ratio is between $$0.5$$ and $$0.55$$ uniformly with respect to $$\varepsilon$$. In fact, this number is close to $$1/\sqrt{\beta} = 1/\sqrt{3}\sim 0.5774$$. Note that, by the product structure of $\( \left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'}^2 = \sum_{T\in{\mathcal{T}}} \left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'(T)}^2,\)$ so that $$\eta({\mathcal{T}})=\sqrt{\beta}\left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'}$$; cf. (2.7). Our numerical results therefore indicate that the slight overestimation of the error by $$\eta$$ is due to the choice of $$\beta$$, and is (asymptotically) uniform in $$\varepsilon$$. Fig. 7. View largeDownload slide Ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$ for the problem from Section 4.4. Fig. 7. View largeDownload slide Ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$ for the problem from Section 4.4. Funding CONICYT through FONDECYT projects (1150056 and 3150012). References Attia ,F. S. , Cai ,Z. & Starke ,G. ( 2009 ) First-order system least squares for the Signorini contact problem in linear elasticity. SIAM J. Numer. Anal. , 47 , 3027 – 3043 . Google Scholar Crossref Search ADS Brezzi ,F. , Hager ,W. W. & Raviart ,P.-A. ( 1977 ) Error estimates for the finite element solution of variational inequalities . Numer. Math. , 28 , 431 – 443 . Google Scholar Crossref Search ADS Brezzi ,F. , Hager ,W. W. & Raviart ,P.-A. ( 1978 ) Error estimates for the finite element solution of variational inequalities. II. Mixed methods . Numer. 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( 2011 ) Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM . ESAIM Math. Model. Numer. Anal. , 45 , 779 – 802 . Google Scholar Crossref Search ADS Glowinski ,R. ( 2008 ) Numerical Methods for Nonlinear Variational Problems . Scientific Computation . Berlin : Springer . Glowinski ,R. , Lions ,J.-L. & Trémolières ,R. ( 1981 ) Numerical Analysis of Variational Inequalities . Studies in Mathematics and its Applications , vol. 8 . Amsterdam : North-Holland . Gopalakrishnan ,J. & Qiu ,W. ( 2014 ) An analysis of the practical DPG method . Math. Comp. , 83 , 537 – 552 . Google Scholar Crossref Search ADS Gwinner ,J. ( 2013 ) $$hp$$-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics . J. Comput. Appl. Math. , 254 , 175 – 184 . Google Scholar Crossref Search ADS Heuer ,N. & Karkulik ,M. ( 2017 ) A Robust DPG method for singularly perturbed reaction–diffusion problems . SIAM J. Numer. 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North-Holland Mathematics Studies, Notas de Matemática [Mathematical Notes] , vol. 134 . Amsterdam : North-Holland , pp. 114 . Scarpini ,F. & Vivaldi ,M. A. ( 1977 ) Error estimates for the approximation of some unilateral problems . RAIRO Anal. Numér. , 11 , 197 – 208 . Google Scholar Crossref Search ADS Signorini ,A. ( 1933 ) Sopra alcune questioni di statica dei sistemi continui . Ann. Sc. Norm. Super. Pisa Cl. Sci.(2) , 2 , 231 – 251 . © 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

On the DPG method for Signorini problems

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Oxford University Press
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Abstract

Abstract We derive and analyse discontinuous Petrov–Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and nonsymmetric formulations, where optimal test functions are used only for the partial differential equation part of the problem, not the boundary conditions. For the symmetric case and lowest-order approximations, we provide a simple a posteriori error estimate. In the second part, we apply our technique to the singularly perturbed case of reaction-dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case. 1. Introduction In this article, we develop a framework to solve contact problems by the discontinuous Petrov–Galerkin method with optimal test functions (DPG method). We consider a simplified model problem ($$-{\it{\Delta}} u+u=f$$ in a bounded domain) with unilateral boundary conditions that resembles a scalar version of the Signorini problem in linear elasticity (Signorini, 1933; Fichera, 1971). We prove well-posedness of our formulation and quasi-optimal convergence. We then illustrate how the scheme can be adapted to the singularly perturbed case of reaction-dominated diffusion ($$-\varepsilon{\it{\Delta}} u + u=f$$). Specifically, we use the DPG setting from Heuer & Karkulik (2017) to design a method that controls the field variables ($$u$$, $$\nabla u$$ and $${\it{\Delta}} u$$) in the so-called balanced norm (which corresponds to $$\|u\|+\varepsilon^{1/4}\|\nabla u\|+\varepsilon^{3/4}\|{\it{\Delta}} u\|$$ with $$L^2$$-norm $$\|\cdot\|$$; Lin & Stynes, 2012). The balanced norm is stronger than the energy norm that stems from the Dirichlet bilinear form of the problem. There is a long history of numerical analysis for contact problems and, more generally, for variational inequalities; cf. the books by Glowinski et al. (1981) and Kikuchi & Oden (1988). Some early articles on the (standard and mixed) finite element method for Signorini-type problems are Hlaváček & Lovíšsek (1977), Brezzi et al. (1977), Scarpini & Vivaldi (1977), Brezzi et al. (1978) and Loppin (1978); see also Gwinner (2013) for a more recent work. Unilateral boundary conditions usually give rise to limited regularity of the solution, and authors have made an effort to establish optimal convergence rates of the finite element method (see, e.g., Chouly & Hild, 2013; Drouet & Hild, 2015). Recently, there has been some interest in extending other numerical schemes to unilateral contact problems, e.g., least squares (Attia et al., 2009), the local discontinuous Galerkin method (Bustinza & Sayas, 2012) and a Nitsche-based finite element method (Chouly & Hild, 2013). Our objective is to set up an appropriate framework to apply DPG technology to variational inequalities, in this case Signorini-type problems. At this point, we do not intend to be more competitive than previous schemes. In particular, we are not concerned with the reduced regularity of solutions that limit convergence orders. In any case, we show well-posedness of our scheme for the minimum regularity of $$u$$ in $$H^1$$ and right-hand-side function $$f$$ in $$L^2$$. Our primary focus is to use DPG schemes in such a way that their performance is not reduced by the presence of Signorini boundary conditions. The DPG method aims at ensuring discrete inf–sup stability by the choice of norms and test functions (cf. Demkowicz & Gopalakrishnan, 2011a,b). This is particularly important for singularly perturbed problems where one can achieve robustness of error control (the discrete inf–sup constant does not depend on perturbation parameters; see Demkowicz & Heuer, 2013; Niemi et al., 2013; Broersen & Stevenson, 2014; Chan et al., 2014; Heuer & Karkulik, 2017). Recently, we have found a setting for the coupling of the DPG method and the Galerkin boundary element method (BEM) to solve transmission problems (see Führer et al., 2017a). The principal idea is to take a variational formulation of the interior problem (suitable for the DPG method) and to add transmission conditions as a constraint. This constraint can be given as boundary integral equations in a least-squares or Galerkin form. In this article, we follow this very strategy. The partial differential equation (PDE) is put in variational form (without considering any boundary condition) and the Signorini conditions are added as a constraint. It turns out that the whole scheme can be written as a variational inequality of the first kind where only the PDE part is tested with optimal test functions, as is the DPG strategy. Then, proving coercivity and boundedness of the bilinear form, the Lions–Stampacchia theorem proves well-posedness. Let us note that there are finite element/boundary element coupling schemes for contact problems (see Carstensen & Gwinner, 1997; Maischak & Stephan, 2005; Gatica et al., 2011). However, their coupling variants generalize the setting of contact problems on bounded domains. In contrast, our coupling scheme (Führer et al., 2017a) (for a standard transmission problem) separates the PDE on the bounded domain from the transmission conditions (and exterior problem) in such a way that they can be formulated as a variational PDE plus constraint. As we have mentioned, this is critical for combining a DPG method with transmission conditions as in Führer et al. (2017a) or with contact conditions as we show here. The singularly perturbed case is more technical for two reasons. First, the DPG setting itself for the PDE is more complicated (we use a robust formulation with three field variables) and, second, the combination of PDE and Signorini conditions has to take into account the diffusion coefficient as scaling parameter. In that way we apply what we have learnt from the DPG scheme (Heuer & Karkulik, 2017) for reaction-dominated diffusion and from the DPG–BEM coupling (Führer & Heuer, 2016) for this case. To the best of our knowledge, this is the first mathematical analysis of a numerical scheme for a singularly perturbed contact problem. We will see that there are symmetric and nonsymmetric forms to include the Signorini boundary conditions. In the symmetric case, we are able to provide a simple a posteriori error estimate that is based on the DPG energy error. Let us also remark that we focus on ultra-weak variational formulations. This has the advantage that both the trace and the flux appear as independent unknowns. This allows for a symmetric formulation of the Signorini conditions, needed for our a posteriori error bound. Other variational formulations can be considered analogously and they give rise to nonsymmetric well-posed formulations. In those cases, however, we have no a posteriori error analysis. Let us also mention that our scheme and analysis of the scalar Signorini problem can be extended to variational inequalities of the second kind, e.g., including Coulomb friction, and to linear elasticity so that, indeed, the Signorini problem can be solved by our DPG scheme. In the following, we continue this introduction by presenting our model problem, by introducing an abstract formulation as a variational inequality of the first kind that is suitable for the DPG scheme and by giving a final overview of the remainder of this article. 1.1 Model problem Let $${\it{\Omega}}\subset{\mathbb{R}}^d$$ ($$d\in\{2,3\}$$) be a simply connected Lipschitz domain with boundary $${\it{\Gamma}}=\partial{\it{\Omega}}$$ and unit normal vector $${\boldsymbol{n}}$$ on $${\it{\Gamma}}$$ pointing towards $${\mathbb{R}}^d\setminus\overline{\it{\Omega}}$$. For $$f\in L^2({\it{\Omega}})$$ we consider the model problem \begin{align}-c{\it{\Delta}} u + u = f \quad\text{in }{\it{\Omega}}, \end{align} (1.1a) subject to the Signorini boundary conditions \begin{align}\label{eq:intro:bc}u \geq 0, \quad \frac{\partial u}{\partial {\boldsymbol{n}}}\geq 0, \quad u \frac{\partial u}{\partial {\boldsymbol{n}}} = 0 \quad\text{on}\quad{\it{\Gamma}}. \end{align} (1.1b) Initially, we will study the case of constant $$c=1$$. Later, we will illustrate the applicability of our DPG scheme to the singularly perturbed problem with constant $$c=\varepsilon$$, assuming $$\varepsilon$$ to be a small positive number. Of course, the restriction of $$u$$ to $${\it{\Gamma}}$$ is understood in the sense of the trace, and its normal derivative on $${\it{\Gamma}}$$ is defined by duality. It is well known that this Signorini problem admits a unique solution $$u\in K := \{ v\in H^1({\it{\Omega}}) \,:\, v \geq 0\ \text{on}\ {\it{\Gamma}} \}$$ (see, e.g., Glowinski et al., 1981; Glowinski, 2008). Moreover, $$u$$ can be characterized as the unique solution of the variational inequality of the first kind \begin{align}\label{eq:intro:weakform}a(u,v-u) \geq (\,f,v-u) \quad\text{for all } v\in K, \end{align} (1.2) with \begin{align*}a(u,v): = {\rm{ }}\left( {\nabla u,\,\nabla v} \right) + {\rm{ }}\left( {u,\,v} \right)\quad {\text{for all }}u, v \in {H^1}({\it{\Omega}} ). \end{align*} In fact, by choosing appropriate test functions $$v\in K$$ and integrating by parts, one finds that problem (1.2) is equivalent to $$u\in K$$, (1.1a) and \begin{align}\label{eq:intro:bcalt}\int_{\it{\Gamma}} \frac{\partial u}{\partial{\boldsymbol{n}}} (v-u) \,{\rm{d}}{\it{\Gamma}} \geq 0 \quad\text{for all } v\in K; \end{align} (1.3) see, e.g., Glowinski et al. (1981). The last relation is useful for establishing a DPG setting of the variational inequality problem. 1.2 Variational formulation Let us give a brief overview of our variational setting for the DPG method. In Section 2, we will consider a nonstandard variational form of (1.1a): \begin{align}\label{eq:intro:dpg}\text{find } {\boldsymbol{u}}\in U \text{ s.t.}\quad b({\boldsymbol{u}},{\boldsymbol{v}}) = L({\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}} \in V. \end{align} (1.4) Here, $$U$$ and $$V$$ are different Hilbert spaces and $$b(\cdot,\cdot):\;U\times V\to{\mathbb{R}}$$ is the bilinear form stemming from, in our case, an ultra-weak formulation. At this point, no boundary conditions are included, so that there is no unique solution to (1.4). Denoting by $$({\cdot},{\cdot})_V$$ the inner product in $$V$$, we define the trial-to-test operator$${\it{\Theta}}_\beta:\;U\to V$$ by \begin{align}\label{eq:dpg:deftttop}{\it{\Theta}}_\beta := \beta{\it{\Theta}} \quad\text{with}\quad({{\it{\Theta}}{\boldsymbol{u}}},{{\boldsymbol{v}}})_V := b({\boldsymbol{u}},{\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}}\in V. \end{align} (1.5) The parameter $$\beta>0$$ has to be selected. Using this operator, the discretization of (1.4) will be based on its equivalent variant with so-called optimal test functions: \begin{align}\label{eq:intro:dpg:theta}\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) = L({\it{\Theta}}_\beta{\boldsymbol{v}}) \quad\text{for all } {\boldsymbol{v}} \in U. \end{align} (1.6) In our formulations, only one component of $${\boldsymbol{u}}\in U$$ corresponds to the original unknown $$u$$. Depending on the particular case, we have to define appropriate Dirichlet and Neumann trace operators $$\gamma_0$$ and $$\gamma_{\boldsymbol{n}}$$ acting on $$U$$. Then, it is left to add the boundary conditions (1.1b) in the form $$\gamma_0 {\boldsymbol{u}}\geq 0$$, $$\gamma_{\boldsymbol{n}} {\boldsymbol{u}} \geq 0$$ and $$\gamma_0{\boldsymbol{u}} \gamma_{\boldsymbol{n}} {\boldsymbol{u}} = 0$$. This transforms (1.6) into a variational inequality. Keeping in mind (1.3), we define the bilinear form \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}) := b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{u}}},{\gamma_0 {\boldsymbol{v}}}\rangle_{\it{\Gamma}} \quad\text{for all } {\boldsymbol{u}},{\boldsymbol{v}} \in U \end{align*} and consider the following formulation: find $${\boldsymbol{u}}\in K^0 := \{ {\boldsymbol{v}}\in U \,:\, \gamma_0{\boldsymbol{v}} \geq 0 \}$$ such that \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all } {\boldsymbol{v}}\in K^0. \end{align*} We will show that this problem is equivalent to (1.1). In particular, it has a unique solution. An intrinsic feature of ultra-weak formulations is that all boundary conditions are essential. Therefore, we can derive methods that use different convex sets. From (1.1b) we infer that \begin{align*}u\left(\psi- \frac{\partial u}{\partial {\boldsymbol{n}}}\right) \geq 0 \quad\text{on }{\it{\Gamma}}\quad \text{for all } \psi\in H^{-1/2}({\it{\Gamma}}) \text{ with } \psi\geq 0, \end{align*} giving rise to the following formulation: find $${\boldsymbol{u}}\in K^{{\boldsymbol{n}}} := \left\{{{\boldsymbol{v}}\in U}:{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}\geq 0} \right\}$$ such that \begin{align*}a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) := b({\boldsymbol{u}},{\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}},{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all } {\boldsymbol{v}}\in K^{\boldsymbol{n}}. \end{align*} Note that neither $$a^0(\cdot,\cdot)$$ nor $$a^{\boldsymbol{n}}(\cdot,\cdot)$$ is symmetric. However, a combination leads to a formulation with symmetric bilinear form as follows: find $${\boldsymbol{u}}\in K^s := \left\{{{\boldsymbol{v}}\in U} \,:\, {\gamma_0{\boldsymbol{v}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{v}}\geq 0} \right\}$$ such that \begin{align*}a^s({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) := \tfrac12(a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}})+a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}})) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) \quad\text{for all }{\boldsymbol{v}}\in K^s. \end{align*} For this symmetric case we will establish a simple a posteriori error estimator. 1.3 Inhomogeneous boundary conditions Instead of (1.1b), the more general boundary conditions \begin{align*}u-g_D \geq 0, \quad \frac{\partial u}{\partial {\boldsymbol{n}}}-g_N\geq 0, \quad(u-g_D) \left(\frac{\partial u}{\partial {\boldsymbol{n}}} -g_N\right)= 0 \quad\text{on}\quad{\it{\Gamma}}, \end{align*} where $$g_{\rm D}$$ denotes the Dirichlet data and $$g_{\rm N}$$ the Neumann data, can be included in our variational setting as follows. As before one observes that \begin{align*}\int_{\it{\Gamma}} \left(\frac{\partial u}{\partial{\boldsymbol{n}}}-g_N\right) (v-u) \,{\rm{d}}{\it{\Gamma}} &\geq 0 \quad\text{for all } v\in H^1({\it{\Omega}}) \text{ with } v|_{\it{\Gamma}}\geq g_D. \end{align*} This gives rise to the following variational formulation: find $${\boldsymbol{u}}\in K^{g_D} := \left\{{{\boldsymbol{v}}\in U} \,:\, {\gamma_0{\boldsymbol{v}}\geq g_D} \right\}$$ s.t. \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq L({\it{\Theta}}_\beta({\boldsymbol{v}}-{\boldsymbol{u}})) + \langle{g_N}, {\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\quad\text{for all }{\boldsymbol{v}}\in K^{g_D}. \end{align*} We note that the bilinear form $$a^0(\cdot,\cdot)$$ is used on the left-hand side. Similarly, one derives formulations using $$a^{\boldsymbol{n}}(\cdot,\cdot)$$, $$a^s(\cdot,\cdot)$$ with different convex sets. We stress that our analysis below holds true in these cases as well. Since, in general, $$K_h^\star\not\subseteq K^\star$$, the a priori analysis differs; see Remark 2.5. For the remainder of this work we stick to the boundary conditions (1.1b). 1.4 Overview The remainder of this article is structured as follows. Section 2 deals with the unperturbed model problem (1.1) (diffusion parameter $$c=1$$). After introducing some notation in Section 2.1, in Section 2.2 we present a variational inequality that represents (1.1) and is based on an ultra-weak variational formulation. Theorem 2.1 then states its well-posedness and equivalence. Afterwards, in Section 2.3 we present the discrete DPG approximation and prove its well-posedness (Theorem 2.2) and quasi-optimal convergence (Theorem 2.3). An a posteriori error estimate is derived in Section 2.4. Some technical results are collected in Section 2.5 and a proof of Theorem 2.1 is given at the end of Section 2. There is a similar structure in Section 3 for the singularly perturbed case (problem (1.1) with small diffusion $$c=\varepsilon$$). Notation is given in Section 3.1 and a variational formulation is presented and analysed in Section 3.2. There, Theorem 3.1 states the well-posedness and equivalence of the variational inequality. Section 3.3 presents and analyses the discrete scheme, its well-posedness (Theorem 3.2), quasi-optimality (Theorem 3.3) and an a posteriori error estimate (Theorem 3.4). Technical results and a proof of Theorem 3.1 are presented in Sections 3.4 and 3.5, respectively. Finally, in Section 4 we present several numerical examples that underline our theoretical results for the unperturbed and singularly perturbed cases. Throughout the article, $$a\lesssim b$$ means that $$a\le kb$$ with a generic constant $$k>0$$ that is independent of involved parameters, functions and the underlying mesh. Similarly, we use the notation $$a\simeq b$$. 2. DPG method 2.1 Notation For Lipschitz domains $$\omega\subset{\mathbb{R}}^d$$, we use the standard Sobolev spaces $$L^2(\omega)$$, $$H^1(\omega)$$ and $${\boldsymbol{H}}({{\rm div\,}},\omega)$$. The $$L^2(\omega)$$ resp. $$L^2(\partial\omega)$$ scalar products are denoted by $$(\cdot ,\cdot)_\omega$$ resp. $$\langle\cdot ,\cdot\rangle_{\partial\omega}$$ with induced norms $$\|\cdot\|_{\omega}$$ resp. $$\|\cdot\|_{{\partial\omega}}$$. Also, we define the trace spaces \begin{align*}H^{1/2}(\partial\omega) := \left\{ \gamma_\omega u\,:\, u\in H^1(\omega) \right\}\quad\text{ and its dual }\quad H^{-1/2}(\partial\omega) := \bigl(H^{1/2}(\partial\omega)\bigr)', \end{align*} where $$\gamma_\omega$$ denotes the trace operator. Here, duality is understood with respect to $$L^2(\partial\omega)$$ as a pivot space, i.e., using the extended $$L^2(\partial\omega)$$ inner product $$\langle{\cdot} ,{\cdot}\rangle_{\partial\omega}$$. The $$L^2({\it{\Omega}})$$ inner product will be denoted by $$({\cdot} ,{\cdot})$$ and the corresponding norm by $$\|\cdot\|_{}$$. Let $${\mathcal{T}}$$ denote a disjoint partition of $${\it{\Omega}}$$ into open Lipschitz sets $${T}\in{\mathcal{T}}$$, i.e., $$\bigcup_{{T}\in{\mathcal{T}}}\overline{T} = \overline{\it{\Omega}}$$. The set of all boundaries of all elements forms the skeleton $${\mathcal{S}} := \left\{ \partial{T} \mid {T}\in{\mathcal{T}} \right\}$$. By $${\boldsymbol{n}}_M$$ we mean the outer normal vector on $$\partial M$$ for a Lipschitz set $$M$$. On a partition $${\mathcal{T}}$$ we use the spaces \begin{align*}H^1({\mathcal{T}}) &:= \{ w\in L^2({\it{\Omega}}) \,:\, w|_T \in H^1(T) \,\forall\, T\in{\mathcal{T}}\}, \\ {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}}) &:= \{ {\boldsymbol{q}}\in (L^2({\it{\Omega}}))^d \,:\, {\boldsymbol{q}}|_T \in {\boldsymbol{H}}({{\rm div\,}},T) \,\forall\, T\in{\mathcal{T}}\}. \end{align*} The symbols $$\nabla_{\mathcal{T}}$$, $${\rm div}_{{\mathcal{T}}}\,$$ resp. $${\Delta}_{{\mathcal{T}}}\,$$ denote the $${\mathcal{T}}$$-piecewise gradient, divergence resp. Laplace operators. On the skeleton $${\mathcal{S}}$$ of $${\mathcal{T}}$$ we introduce the trace spaces \begin{align*}H^{1/2}({\mathcal{S}}) &:= \Big\{ \widehat u \in {\it{\Pi}}_{{T}\in{\mathcal{T}}}H^{1/2}(\partial{T})\,:\, \exists\, w\in H^1({\it{\Omega}}) \text{ such that }\widehat u|_{\partial{T}} = w|_{\partial{T}}\; \forall\, {T}\in{\mathcal{T}} \Big\},\\ H^{-1/2}({\mathcal{S}}) &:= \Big\{ \widehat\sigma \in {\it{\Pi}}_{{T}\in{\mathcal{T}}}H^{-1/2}(\partial{T})\,:\, \exists\, {\boldsymbol{q}}\in{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}) \text{ such that }\widehat\sigma|_{\partial{T}} = ({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \Big\}. \end{align*} These spaces are equipped with norms depending on the problem; see Sections 2.2 and 3.1. For functions $$\widehat u\in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma\in H^{-1/2}({\mathcal{S}})$$ and $${\boldsymbol\tau}\in{\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}})$$, $$v\in H^1({\mathcal{T}})$$ we define \begin{align*}\langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}}:= \sum_{{T}\in{\mathcal{T}}}\langle{\widehat u|_{\partial{T}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{T}}\rangle_{\partial{T}},\quad\langle{\widehat\sigma} ,{v}\rangle_{\mathcal{S}}:= \sum_{{T}\in{\mathcal{T}}}\langle{\widehat\sigma|_{\partial{T}}} ,{v}\rangle_{\partial{T}}. \end{align*} With the latter relations we can also define the restrictions $$\widehat u|_{\it{\Gamma}} \in H^{1/2}({\it{\Gamma}})$$ and $$\widehat \sigma|_{\it{\Gamma}} \in H^{-1/2}({\it{\Gamma}})$$ of functions $$\widehat u\in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma \in H^{-1/2}({\mathcal{S}})$$ onto $${\it{\Gamma}}$$. Let $$w\in H^1({\it{\Omega}})$$ be such that $$w|_{\partial T} = \widehat u|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. Then, \begin{align} \label{trace}\langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= \sum_{T\in{\mathcal{T}}}\langle{w|_{\partial T}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_T}\rangle_{\partial T} = \sum_{T\in{\mathcal{T}}} ({\nabla w} ,{{\boldsymbol\tau}})_T + ({w} ,{{{\rm div\,}} {\boldsymbol\tau}})_T = ({\nabla w} ,{{\boldsymbol\tau}}) + ({w} ,{{{\rm div\,}} {\boldsymbol\tau}}) \nonumber\\ &= \langle{w|_{\it{\Gamma}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}} =: \langle{\widehat u|_{\it{\Gamma}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}}\quad\text{for all }{\boldsymbol\tau}\in{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}). \end{align} (2.1) Note that $$\widehat u|_{\it{\Gamma}}$$ is uniquely determined since the above relation is independent of the choice of the extension $$w\in H^1({\it{\Omega}})$$ and since $$H^{-1/2}({\it{\Gamma}})$$ is the (normal) trace space of $${\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$. Similarly, we define $$\widehat\sigma|_{\it{\Gamma}} \in H^{-1/2}({\it{\Gamma}})$$ through \begin{align} \label{ntrace}\langle{\widehat \sigma} ,{v}\rangle_{\mathcal{S}} &= \sum_{T \in{\mathcal{T}}}\langle{{\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{T}} ,{v|_{\it{\Gamma}}}\rangle_{\partial T} = \sum_{T\in{\mathcal{T}}} ({{{\rm div\,}} {\boldsymbol\sigma}} ,{v})_T + ({{\boldsymbol\sigma}} ,{\nabla v})_T = ({{{\rm div\,}} {\boldsymbol\sigma}} ,{v}) + ({{\boldsymbol\sigma}} ,{\nabla v}) \nonumber\\ &= \langle{{\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{v|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} =: \langle{\widehat\sigma|_{\it{\Gamma}}} ,{v|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} \quad\text{for all } v\in H^1({\it{\Omega}}), \end{align} (2.2) where $${\boldsymbol\sigma}\in {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ with $${\boldsymbol\sigma}\cdot{\boldsymbol{n}}_{T}|_{\partial T} = \widehat\sigma|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. 2.2 Ultra-weak variational formulation We derive an ultra-weak formulation of (1.1a) with $$c=1$$. Following Demkowicz & Gopalakrishnan (2011a), we define $${\boldsymbol\sigma} = \nabla u$$. Then, \begin{align*}-{{\rm div\,}}{\boldsymbol\sigma} + u = f, \quad {\boldsymbol\sigma} -\nabla u = 0. \end{align*} We define $$\widehat u \in H^{1/2}({\mathcal{S}})$$ and $$\widehat\sigma\in H^{-1/2}({\it{\Gamma}})$$ such that $$\widehat u|_{\partial T} = u|_{\partial T}$$ and $$\widehat\sigma|_{\partial T} = \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$. Testing the first-order system with functions $$v\in H^1({\mathcal{T}})$$, $${\boldsymbol\tau}\in {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}})$$, and integrating by parts, we end up with the ultra-weak formulation \begin{align}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) + ({u} ,v) - \langle{\widehat\sigma} ,{v}\rangle_{{\mathcal{S}}} &= ({f} ,v), \\ \end{align} (2.3a) \begin{align}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{{\mathcal{S}}} &= 0. \end{align} (2.3b) This gives rise to a bilinear form $$b:U \times V \to {\mathbb{R}}$$ and functional $$L:V\to {\mathbb{R}}$$ defined by \begin{align*}b({\boldsymbol{u}},{\boldsymbol{v}}) &:= ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) + ({u} ,v) - \langle{\widehat\sigma} ,{v}\rangle_{{\mathcal{S}}}+ ({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{{\mathcal{S}}}, \\ L({\boldsymbol{v}}) &:= ({f} ,v) \end{align*} for all $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma) \in U$$, $${\boldsymbol{v}} = (v,{\boldsymbol\tau}) \in V$$, where \begin{align*}U &:= L^2({\it{\Omega}})\times [L^2({\it{\Omega}})]^d \times H^{1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}), \\ V &:= H^1({\mathcal{T}})\times {\boldsymbol{H}}({{\rm div\,}},{\mathcal{T}}). \end{align*} We equip these spaces with the norms \begin{align*}\|{{\boldsymbol{u}}}\|{_{U}^{2}} &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \|{\widehat u}\|_{1/2,{\mathcal{S}}}^2 +\|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}}^2, \\ \|{{\boldsymbol{v}}}\|_{V}^2 &:= \|{v}\|_{}^2 + \|{\nabla_{\mathcal{T}} v}\|_{}^2 + \|{{\boldsymbol\tau}}\|_{}^2 + \|{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}\|_{}^2, \end{align*} where the norms for the trace variables are given by the image norms (minimum energy extensions to $$H^1({\it{\Omega}})$$ resp. $${\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$) \begin{align*}\|{\widehat u}\|_{1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), \widehat u|_{\partial{T}}=w|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\|{{{\rm div\,}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}),\ \widehat\sigma|_{\partial{T}}\!=\!({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\!\in\!{\mathcal{T}} \right\}. \end{align*} In $$H^{1/2}({\it{\Gamma}})$$, we use the norm \begin{align*}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})} := \inf \left\{ (\|{w}\|_{}^2 + \|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}),\widehat v = w|_{\it{\Gamma}} \right\}, \end{align*} and define the norm $$\|{\cdot}\|_{H^{-1/2}({\it{\Gamma}})}$$ by duality. The bilinear form $$b(\cdot,\cdot)$$ induces a linear operator $$B:\;U\to V'$$ so that (2.3) can be written as \[\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad B{\boldsymbol{u}} = L. \] The operator $$B$$ has a nontrivial kernel. To consider the boundary conditions, we define trace operators \begin{align*}\begin{aligned}\gamma_0&:\;U\to H^{1/2}({\it{\Gamma}}),&\quad& \gamma_0(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma):=\widehat u|_{\it{\Gamma}}&\quad&\text{(Dirichlet trace, see (2.1)),} \\ \gamma_{\boldsymbol{n}}&:\;U\to H^{-1/2}({\it{\Gamma}}),&\quad& \gamma_{\boldsymbol{n}}(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma):=\widehat \sigma|_{\it{\Gamma}}&\quad&\text{(Neumann trace, see (2.2))}, \end{aligned}\end{align*} and define the sets \begin{align*}K^0 := \left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}} \geq 0} \right\}, \quad K^{\boldsymbol{n}} :=\left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_{\boldsymbol{n}}{\boldsymbol{u}}\geq 0} \right\}, \quad K^s := \left\{{{\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{u}}\geq 0} \right\}. \end{align*} The relations ‘$$\geq$$’ are partial orderings. Following Kikuchi & Oden (1988, Section 5) we define \begin{align*}\widehat v \geq 0 \text{ in } H^{1/2}({\it{\Gamma}}) \quad :\Longleftrightarrow \quad\exists\, \{\widehat v_n\}_{n\in {\mathbb{N}}} \text{ s.t. } v_n \in \mathrm{Lip}({\it{\Gamma}}) \text{ with }v_n\geq 0 \text{ and } v_n \rightharpoonup v \quad\text{in } H^{1/2}({\it{\Gamma}}), \end{align*} where $$\mathrm{Lip}({\it{\Gamma}})$$ denotes all Lipschitz continuous functions $${\it{\Gamma}} \to {\mathbb{R}}$$. We note that this definition of $$\widehat v\geq 0$$ is equivalent to $$\widehat v\geq 0$$ a.e. (see Kikuchi & Oden, 1988, Section 5). On $$H^{-1/2}({\it{\Gamma}})$$, ‘$$\geq$$’ is understood as duality (see, e.g., Hlaváček et al., 1988, Section 1.1.11), \begin{align*}\widehat\sigma \geq 0 \text{ in } H^{-1/2}({\it{\Gamma}}) \quad:\Longleftrightarrow \quad\langle{\widehat\sigma} ,{\widehat v}\rangle_{\it{\Gamma}} \geq 0 \quad\text{for all } \widehat v\in H^{1/2}({\it{\Gamma}}) \text{ with } \widehat v\geq 0. \end{align*} One can verify that $$\left\{{\widehat v\in H^{1/2}({\it{\Gamma}})} \,:\, {\widehat v\geq 0} \right\}$$ and $$\left\{{\widehat\sigma\in H^{-1/2}({\it{\Gamma}})} \,:\, {\widehat\sigma\geq 0} \right\}$$ are closed, convex sets. We will also see that $$K^0$$, $$K^{\boldsymbol{n}}$$ and $$K^s$$ are nonempty, closed, convex subsets of $$U$$ (see Lemma 2.8 below). Now let $$u\in H^1({\it{\Omega}})$$ denote the solution of problem (1.1) and define $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in U$$ with components as above. Then $$B{\boldsymbol{u}}=L$$ and, considering inequality (1.3) as a representation of the boundary conditions (1.1b), one sees that $${\boldsymbol{u}}\in K^\star$$ for $$\star\in\{0,{\boldsymbol{n}},s\}$$ and \begin{align*}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\geq 0 \quad\text{for all }{\boldsymbol{v}}\in K^0, \\ \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\geq 0 \quad\text{for all } {\boldsymbol{v}}\in K^{\boldsymbol{n}}, \\ \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}) &\geq 0 \quad\text{for all } {\boldsymbol{v}}\in K^s. \end{align*} Recalling the formulation (1.6) of the problem $$B{\boldsymbol{u}}=L$$, this leads us to defining the bilinear forms $$a^\star:U\times U\to {\mathbb{R}}$$, $$\star\in\{0,{\boldsymbol{n}},s\}$$ and the functional $$F:U\to{\mathbb{R}}$$ as \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}, \\ a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}, \\ a^s({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}), \\ F({\boldsymbol{v}}) &:= L({\it{\Theta}}_\beta{\boldsymbol{v}}) \end{align*} for $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. Here, $$\beta>0$$ is a constant that will be fixed below. We then obtain the following formulations of the model problem (1.1) (with $$c=1$$) as ultra-weak variational inequalities: for fixed $$\star\in\{0,{\boldsymbol{n}},s\}$$ find $${\boldsymbol{u}}\in K^\star$$ such that \begin{align}\label{eq:dpg:varineq}a^\star({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq F({\boldsymbol{v}}-{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{v}}\in K^\star. \end{align} (2.4) These are variational inequalities of the first kind, and we use a standard framework for their analysis (cf. Glowinski et al., 1981; Glowinski, 2008). The following is one of our main results. Theorem 2.1 Fix $$\star\in\{0,{\boldsymbol{n}},s\}$$. For all $$\beta\geq 2$$ the following holds: the bilinear form $$a^\star:U\times U \to {\mathbb{R}}$$ is $$U$$-coercive, \begin{align*}\|{{\boldsymbol{u}}}\|_{U}^2 \leq C_1 a^\star({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{u}}\in U, \end{align*} and bounded, \begin{align*}|a^\star({\boldsymbol{u}},{\boldsymbol{v}})| \leq (C_2^2 \beta+1) \|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U} \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}} \in U. \end{align*} The constant $$C_1>0$$ depends only on $${\it{\Omega}}$$ and $$C_2=\|{b}\|_{}=\|{B}\|_{}$$. In particular, the variational inequality (2.4) is uniquely solvable and equivalent to problem (1.1) with $$c=1$$ in the following sense. If $$u\in H^1({\it{\Omega}})$$ solves problem (1.1), then $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in K^\star$$ with $${\boldsymbol\sigma} := \nabla u$$, $$\widehat u|_{\partial T} := u|_{\partial T}$$, $$\widehat\sigma|_{\partial T} := \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$ solves (2.4). On the other hand, if $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in K^\star$$ solves (2.4), then $$u\in H^1({\it{\Omega}})$$ solves (1.1). Moreover, the unique solution $${\boldsymbol{u}}\in K^\star$$ of (2.4) satisfies \begin{align}\label{eq:dpg:exactsol}b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{w}}) = F({\boldsymbol{w}}) \quad\text{for all } {\boldsymbol{w}}\in U. \end{align} (2.5) This theorem is proved in Section 2.6. 2.3 Discretization and convergence To discretize our variational inequality (2.4), we use, in this work, lowest-order piecewise polynomial functions. That is, we replace the space $$U$$ by \begin{align*}U_h := P^0({\mathcal{T}})\times [P^0({\mathcal{T}})]^d \times S^1({\mathcal{S}}) \times P^0({\mathcal{S}}), \end{align*} where $$P^0$$ denotes the space of elementwise constants on $${\mathcal{T}}$$ resp. $${\mathcal{S}}$$, and $$S^1({\mathcal{S}})$$ is the space of globally continuous, elementwise affine functions on $${\mathcal{S}}$$. Defining the nonempty convex subsets \begin{align*}K_h^0 &:= \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0} \right\}, \\ K_h^{\boldsymbol{n}} &:= \left\{{{\boldsymbol{v}}_h\in U_h}:{\gamma_{\boldsymbol{n}} {\boldsymbol{v}}_h\geq 0} \right\}, \\ K_h^s &:= \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h\geq 0} \right\}, \end{align*} we find that $$K_h^\star\subseteq K^\star$$. The discretized version of (2.4) then reads, for fixed $$\star\in\{0,{\boldsymbol{n}},s\}$$ find $${\boldsymbol{u}}_h\in K_h^\star$$ such that \begin{align}\label{eq:dpg:varineqdisc}a^\star({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h)\geq F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \quad\text{for all } {\boldsymbol{v}}_h\in K_h^\star. \end{align} (2.6) Coercivity and boundedness of $$a^\star(\cdot,\cdot)$$ hold on the full space $$U$$ (Theorem 2.1). Therefore, the Lions–Stampacchia theorem applies also in the discrete case. Theorem 2.2 Under the assumptions of Theorem 2.1, the discrete variational inequality (2.6) admits a unique solution $${\boldsymbol{u}}_h\in K_h^\star$$. Our scheme converges quasi-optimally in the following sense. Theorem 2.3 For $$\beta\ge 2$$ let $${\boldsymbol{u}}\in K^\star$$, $${\boldsymbol{u}}_h\in K_h^\star$$ denote the exact solutions of (2.4), (2.6). Then it holds that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \begin{cases}\inf_{{\boldsymbol{v}}_h\in K_h^0} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 +\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}\right) & (\star = 0), \\ \inf_{{\boldsymbol{v}}_h\in K_h^{\boldsymbol{n}}} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 +\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\right) & (\star = {\boldsymbol{n}}), \\ \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 + \tfrac12(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right) & (\star = s). \end{cases}\end{align*} The generic constant depends on $${\it{\Omega}}$$ and $$\beta$$ but not on $${\mathcal{T}}$$. Proof. By Theorem 2.1, $$a^\star(\cdot,\cdot)$$ is $$U$$-coercive and bounded. Therefore, we can follow Falk’s lemma (Falk, 1974) to deduce that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2 &\lesssim a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{u}}_h) = a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{v}}_h) + a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ & \leq \|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U (C_2^2\beta+1) \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U + a^\star({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \end{align*} for all $${\boldsymbol{v}}_h\in K_h^\star\subseteq K^\star$$. We consider only the case $$\star = 0$$. The remaining cases are treated in the same manner. To tackle the last term on the right-hand side, we use (2.5) and (2.6) to see that \begin{align*}a^0({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) &= b({\boldsymbol{u}},{\it{\Theta}}_\beta ({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) ) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}- a^0({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ &= \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}} + F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)-a^0({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \\ & \leq \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}. \end{align*} Note that the exact solution $${\boldsymbol{u}}$$ satisfies $$\gamma_{\boldsymbol{n}} {\boldsymbol{u}} \geq 0$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = 0$$. For the discrete one it holds that $$\gamma_0{\boldsymbol{u}}_h\geq 0$$. Hence, the last term further simplifies to \begin{align*}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}} \leq \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0 {\boldsymbol{v}}_h}\rangle_{\it{\Gamma}}= \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}. \end{align*} Altogether, Young’s inequality with some parameter $$\delta>0$$ shows that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2\lesssim \frac{\delta}2 \|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U}^2 + \frac{\delta^{-1}}2 (C_2^2\beta+1)^2\|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}\end{align*} for arbitrary $${\boldsymbol{v}}_h\in K_h^0$$. This proves the a priori estimate. □ Let $${\mathcal{S}}_{\it{\Gamma}} := \left\{{\partial T \cap {\it{\Gamma}}} \,:\, {T\in{\mathcal{T}}} \right\}$$ denote the mesh on the boundary, which is induced by the volume mesh $${\mathcal{T}}$$. Remark 2.4 To deduce convergence rates assume, for instance, that $$u\in H^3({\it{\Omega}})$$ is the solution of (1.1). Set \begin{align*}{\boldsymbol{v}}_h := ({\it{\Pi}}_h^0 u,\boldsymbol{{\it{\Pi}}}_h^0\nabla u,I_h u|_{\mathcal{S}}, ({\it{\Pi}}_h^{{{\rm div\,}}} \nabla u\cdot{\boldsymbol{n}}_T|_{\partial T})_{T\in{\mathcal{T}}}) \in U, \end{align*} where $${\it{\Pi}}_h^0$$ and $$\boldsymbol{{\it{\Pi}}}_h^0$$ are the $$L^2({\it{\Omega}})$$-orthogonal projections onto the elementwise constant spaces, $$I_h$$ is the nodal interpolant and $${\it{\Pi}}_h^{{{\rm div\,}}}$$ is the (lowest-order) Raviart–Thomas projector. Note that $$I_h$$ preserves non-negativity (in particular, on the boundary) and the normal trace of $${\it{\Pi}}_h^{{{\rm div\,}}}$$ is the $$L^2({\it{\Gamma}})$$-orthogonal projection $$\pi_h^0$$ and, hence, preserves non-negativity on the boundary as well. Thus, $${\boldsymbol{v}}_h \in K_h^\star$$ for $$\star\in\{0,{\boldsymbol{n}},s\}$$. We refer to Demkowicz & Gopalakrishnan (2011a) to see that $$\|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U = {\mathcal O}(h)$$. To control the boundary terms, we note that \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}_h-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| \leq \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{{\it{\Gamma}}}\|{(u-I_h u)|_{\it{\Gamma}}}\|_{{\it{\Gamma}}} = {\mathcal{O}}(h^2). \end{align*} Let $${\it{\Gamma}}_1,\dots,{\it{\Gamma}}_L \subseteq {\it{\Gamma}}$$ be such that $$\bigcup \overline{\it{\Gamma}}_j = {\it{\Gamma}}$$ and $${\boldsymbol{n}}_{\it{\Omega}}|_{{\it{\Gamma}}_j}$$ is constant. Using the projection properties of $${\it{\Pi}}_h^{{{\rm div\,}}}$$ and $$\pi_h^0$$ and the local approximation property of $$\pi_h^0$$ we obtain \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| &\leq \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_E| = \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}-\pi_h^0\gamma_0{\boldsymbol{u}}}\rangle_E| \\ & \lesssim \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} |E| \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^1(E)} |E| \|{\gamma_0{\boldsymbol{u}}}\|_{H^1(E)}\lesssim h^2 \left(\sum_{j=1}^L \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^1({\it{\Gamma}}_j)}^2\right)^{1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^1({\it{\Gamma}})}. \end{align*} Altogether, using $${\boldsymbol{v}}_h$$ in Theorem 2.3, we infer \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 &\lesssim \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_U^2 + \begin{cases}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}| & (\star = 0) \\ |\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| & (\star = {\boldsymbol{n}}) \\ \tfrac12 |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}| & (\star = s) \end{cases}\\&= {\mathcal O}(h^2). \end{align*} With less regularity of $$u$$ the treatment of the boundary terms becomes more technical. We refer to Drouet & Hild (2015) for details. Remark 2.5 Our analysis also allows for nonconforming discrete cones $$K_h^\star\not\subseteq K^\star$$. Then, an additional consistency error shows up in Theorem 2.3, i.e., in the case $$\star = 0$$, \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^0} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}_h-\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \right) + \inf_{{\boldsymbol{v}}\in K^0}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}-\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}. \end{align*} Again, this a priori estimate can be derived using Falk’s lemma. 2.4 A posteriori error estimate We derive a simple error estimator in the symmetric case, i.e., $$\star=s$$. Throughout this section we assume that $$\beta>0$$ is a fixed constant, such that $$a^s(\cdot,\cdot)$$ is coercive (Theorem 2.1). Let $${\boldsymbol{u}}_h\in K_h^s$$ denote the unique solution of (2.6). We define for all $$T\in{\mathcal{T}}$$, local volume error indicators $$\eta(T)$$ as the norm of the residual $$L-B{\boldsymbol{u}}_h$$ restricted to $$T$$, or, formally, \begin{align} \label{eq:aposteriori:estdef}\eta(T)^2 := \beta \|{R_T^{-1}\iota_T^*(L-B{\boldsymbol{u}}_h)}\|_{V(T)}^2. \end{align} (2.7) Here, $$V(T) := H^1(T) \times {\boldsymbol{H}}({{\rm div\,}},T)$$, $$\|{\cdot}\|_{V(T)}$$ denotes the canonical norm on $$V(T)$$, $$R_T:\;V(T) \to (V(T))'$$ is the Riesz isomorphism, and $$\iota_T^*$$ is the dual of the canonical embedding $$\iota_T:\; V(T)\to V$$. Moreover, for all $$E\in{\mathcal{S}}_{\it{\Gamma}}$$ we define local boundary indicators by \begin{align*}\eta(E)^2 := \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_E. \end{align*} Note that $${\boldsymbol{u}}_h\in K_h^s$$ implies that $$\eta(E)^2\geq 0$$. The overall estimator is then given by \begin{align*}\eta^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} Theorem 2.6 For $$\beta\ge 2$$ let $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ be the solutions of (2.4) and (2.6), respectively. Then, there holds the reliability estimate \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U \leq C_\mathrm{rel} \eta, \end{align*} with a constant $$C_\mathrm{rel}>0$$ that depends on $${\it{\Omega}}$$ but not on $${\mathcal{T}}$$ or $$\beta$$. Proof. By the $$U$$-coercivity of $$a^s(\cdot,\cdot)$$ (see Theorem 2.1) we have \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 &\lesssim a^s({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\boldsymbol{u}}-{\boldsymbol{u}}_h) = b({\boldsymbol{u}}-{\boldsymbol{u}}_h,{\it{\Theta}}_\beta({\boldsymbol{u}}-{\boldsymbol{u}}_h) ) + \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-{\boldsymbol{u}}_h)} ,{\gamma_0({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}\\ &=\beta \|{B({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-{\boldsymbol{u}}_h)} ,{\gamma_0({\boldsymbol{u}}-{\boldsymbol{u}}_h)}\rangle_{\it{\Gamma}}. \end{align*} Note that $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ imply that $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}\geq 0$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\geq 0$$. Together with $$B{\boldsymbol{u}} = L$$ and $$\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = 0$$ we obtain the estimate \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_U^2 \lesssim \beta \|{L-B{\boldsymbol{u}}_h}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}} {\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_{\it{\Gamma}}= \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2, \end{align*} which finishes the proof. □ 2.5 Technical details We start by proving boundedness of our trace operators that are specifically modified for the space $$U$$. Lemma 2.7 The operators $$\gamma_0 : (U,\|\cdot\|_{U})\to (H^{1/2}({\it{\Gamma}}),\|\cdot\|_{H^{1/2}({\it{\Gamma}})})$$ and $$\gamma_{\boldsymbol{n}} : (U,\|\cdot\|_{U})\to (H^{-1/2}({\it{\Gamma}}), \|\cdot\|_{H^{-1/2}({\it{\Gamma}})})$$ have unit norm. Proof. Boundedness of these operators follows basically by definition of the corresponding norms (see also Führer et al., 2017a, Lemma 3). Specifically, the definitions of the norms $$\|\cdot\|_{H^{1/2}({\it{\Gamma}})}$$, $$\|\cdot\|_{1/2,{\mathcal{S}}}$$ and $$\|\cdot\|_{U}$$ prove boundedness of $$\gamma_0$$ with bound $$1$$. That the norm of these operators is equal to $$1$$ follows by considering appropriate extensions. Note that $$\|{\cdot|_{\it{\Gamma}}}\|_{H^{-1/2}({\it{\Gamma}})} \leq \|{\cdot}\|_{-1/2,{\mathcal{S}}}$$ and, thus, the definition of $$\|\cdot\|_{U}$$ shows boundedness of $$\gamma_{\boldsymbol{n}}$$. □ By the boundedness of the operators $$\gamma_0$$ and $$\gamma_{\boldsymbol{n}}$$ we immediately establish the following result. Lemma 2.8 The sets $$K^\star$$ ($$\star\in\{0,{\boldsymbol{n}},s\}$$) are nonempty, closed, convex subsets of $$U$$. The following steps are to characterize the kernel of the operator $$B$$. This kernel is nontrivial since $$B$$ does not include any boundary condition. Our procedure is similar to the one in Führer et al. (2017a). For a function $$v\in H^{1/2}({\it{\Gamma}})$$, we define its quasi-harmonic extension $$\widetilde u\in H^1({\it{\Omega}})$$ as the unique solution of \begin{align}\label{eq:dpg:harmext}-{\it{\Delta}} \widetilde u + \widetilde u &=0 \quad\text{in }{\it{\Omega}}, \quad\widetilde u|_{\it{\Gamma}} = v. \end{align} (2.8) Note that the infimum in the definition of $$\|{v}\|_{H^{1/2}({\it{\Gamma}})}$$ is attained for the function $$\widetilde u$$, i.e., $$\|{\widetilde u}\|_{}^2 + \|{\nabla\widetilde u}\|_{}^2 = \|{v}\|_{H^{1/2}({\it{\Gamma}})}^2$$. Then, define the operator $$\mathcal{E} : H^{1/2}({\it{\Gamma}})\to U$$ by \begin{align*}\mathcal{E} v := (\widetilde u,{\boldsymbol\sigma},\widehat u,\widehat\sigma), \text{ where }{\boldsymbol\sigma} := \nabla\widetilde u, \quad \widehat u|_{\partial T} := \widetilde u|_{\partial T}, \quad \widehat\sigma|_{\partial T} := \nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T} \quad\text{for all } T\in{\mathcal{T}}. \end{align*} We combine important properties of $$B$$ and $$\mathcal{E}$$ in the following lemma. Lemma 2.9 The operators $$B : U \to V'$$, $$\mathcal{E}: H^{1/2}({\it{\Gamma}})\to U$$ have the following properties: (i) The operators $$B$$ and $$\mathcal{E}$$ are bounded. Specifically it holds that \begin{align*}&\|{B{\boldsymbol{u}}}\|_{V'} \lesssim \|{{\boldsymbol{u}}}\|_{U} &&\quad\text{for all } {\boldsymbol{u}}\in U,\\ &\|{v}\|_{H^{1/2}({\it{\Gamma}})} \le \|{\mathcal{E} v}\|_{U} \le \sqrt{3} \|{v}\|_{H^{1/2}({\it{\Gamma}})}&&\quad\text{for all } v \in H^{1/2}({\it{\Gamma}}). \end{align*} (ii) The kernel of $$B$$ consists of all quasi-harmonic extensions, i.e., $$\ker(B) = \operatorname{ran}(\mathcal{E})$$. (iii) The operator $$\mathcal{E}$$ is a right inverse of $$\gamma_0$$. (iv) The operator $$B:\;U/\ker(B)\to V'$$ is inf–sup stable, \begin{align*}\|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U} \lesssim \|{B{\boldsymbol{u}}}\|_{V'} = \sup_{{\boldsymbol{v}}\in V} \frac{\langle{B{\boldsymbol{u}}} ,{{\boldsymbol{v}}}\rangle}{\|{{\boldsymbol{v}}}\|_{V}}= b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align*} The involved constant depends only on $${\it{\Omega}}$$. Proof. In the case of the Poisson equation, these results have been established in Führer et al. (2017a). In a recent work (Führer et al., 2017b), we analysed a time-stepping scheme for the heat equation, which naturally leads to the equation $$-{\it{\Delta}} u + \delta^{-1}u = f$$, where $$\delta$$ corresponds to a time step $$k_n$$. Setting $$\delta=k_n=1$$ in Führer et al. (2017b, Lemma 8), we obtain boundedness of the operator $$B$$ and stability \begin{align*}\|{{\boldsymbol{u}}_0}\|_{U} \lesssim \|{B{\boldsymbol{u}}_0}\|_{V'} \quad\text{for all } {\boldsymbol{u}}_0 \in U \text{ with } \gamma_0{\boldsymbol{u}}_0 = 0. \end{align*} By definition of $$\mathcal{E}$$ one sees (iii). Furthermore, integration by parts shows $$\operatorname{ran}(\mathcal{E})\subseteq \ker(B)$$. For $${\boldsymbol{u}}\in U$$ we define $${\boldsymbol{u}}_0:={\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}$$ and infer \begin{align*}\|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_U \lesssim \|{B({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})}\|_{V'} = \|{B{\boldsymbol{u}}}\|_{V'}, \end{align*} which proves (iv) as well as $$\ker(B)\subseteq \operatorname{ran}(\mathcal{E})$$, hence (ii). It remains to show the relations for $$\mathcal{E}$$ in (i). Let $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\widehat u,\widehat\sigma) = \mathcal{E} v$$ for $$v\in H^{1/2}({\it{\Gamma}})$$. Then, \begin{align*}\|{\mathcal{E} v}\|_U^2 \geq \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 = \|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 = \|{v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} On the other hand, using that $${\it{\Delta}} u = u$$ by construction of $$\mathcal{E} v$$, we deduce that \begin{align*}\|{\mathcal{E} v}\|_U^2 &= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \|{\widehat u}\|_{1/2,{\mathcal{S}}}^2 + \|{\widehat\sigma}\|_{-1/2,{\it{\Gamma}}}^2 \\ &\leq \|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 +\|{u}\|_{}^2 + \|{\nabla u}\|_{}^2 + \|{\nabla u}\|_{}^2 + \|{{\it{\Delta}} u}\|_{}^2 \\ &= 3\|{u}\|_{}^2 + 3\|{\nabla u}\|_{}^2 = 3\|{v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} This concludes the proof. □ In Lemma 2.11 below, we give an explicit bound for the control of the Neumann trace for functions of the quotient space $$U/\ker(B)$$. For its proof we need the following technical result. Lemma 2.10 Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$. The problem \begin{align}{{\rm div\,}}{\boldsymbol\tau} + v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:dpg:dualestimate:a} \\ \end{align} (2.9a) \begin{align}{\boldsymbol\tau} + \nabla v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:dpg:dualestimate:b} \\ \end{align} (2.9b) \begin{align}v|_{{\it{\Gamma}}} &= \widehat v \label{eq:dpg:dualestimate:c}\end{align} (2.9c) admits a unique solution $$(v,{\boldsymbol\tau}) \in H^1({\it{\Omega}})\times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ with $${\it{\Delta}} v\in H^1({\it{\Omega}})$$ and \begin{align*}\|{{\boldsymbol\tau}}\|_{}^2 + \|{{{\rm div\,}}{\boldsymbol\tau}}\|_{}^2 = \|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 = \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Proof. Let $$v\in H^1({\it{\Omega}})$$ be the unique solution of \begin{align*}-{\it{\Delta}} v + v = 0\quad\text{in } {\it{\Omega}}, \quad v|_{\it{\Gamma}} = \widehat v. \end{align*} Then, $$\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 = \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2$$ by definition of the latter norm. Define $${\boldsymbol\tau}:=-\nabla v\in L^2({\it{\Omega}})$$. Since $${\it{\Delta}} v = v \in H^1({\it{\Omega}})$$, we have $${\boldsymbol\tau}\in {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$, and $${{\rm div\,}}{\boldsymbol\tau} = -{\it{\Delta}} v = -v$$ shows (2.9a). To see unique solvability, let additionally $$(v_2,{\boldsymbol\tau}_2)\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}})$$ solve (2.9). The difference $$w:=v-v_2$$ satisfies $$-{\it{\Delta}} w + w =0$$ in $${\it{\Omega}}$$ with $$w|_{\it{\Gamma}} = 0$$. Thus, $$w=0$$ and $${\boldsymbol\tau} = -\nabla v = -\nabla v_2 = {\boldsymbol\tau}_2$$ as well. □ Lemma 2.11 It holds that \begin{align}\label{eq:dpg:Hm12estimate}\|{\gamma_{\boldsymbol{n}}\|({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})}\|_{H^{-1/2}({\it{\Gamma}})} \leq \sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align} (2.10) Proof. Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ and choose the test function $${\boldsymbol{v}}=(v,{\boldsymbol\tau})\in H^1({\it{\Omega}}) \times {\boldsymbol{H}}({{\rm div\,}},{\it{\Omega}}) \subseteq V$$ to be the solution of (2.9). By Lemma 2.10 it holds $$\|{{\boldsymbol{v}}}\|_{V}=\sqrt{2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}$$. Then, by the definition of the bilinear form $$b(\cdot,\cdot)$$ and (2.1), (2.2) we have $$b({\boldsymbol{u}},{\boldsymbol{v}}) = -\langle{\widehat\sigma} ,{v}\rangle_{\mathcal{S}} -\langle{\widehat u} ,{\tau\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} = -\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\widehat v}\rangle_{\it{\Gamma}} - \langle{\gamma_0{\boldsymbol{u}}} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}_{\it{\Omega}}}\rangle_{\it{\Gamma}}$$. Since $$\gamma_0({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}) = 0$$ it follows that \begin{align*}\lvert \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\widehat v}\rangle_{\it{\Gamma}} \rvert &= \lvert b({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}},{\boldsymbol{v}})\rvert = \lvert b({\boldsymbol{u}},{\boldsymbol{v}})\rvert \leq \|{B{\boldsymbol{u}}}\|_{V'} \|{{\boldsymbol{v}}}\|_{V}= \sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}, \end{align*} where we have used that $$\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}\in \ker(B)$$. Dividing by $$\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}$$ and taking the supremum over all $$\widehat v\in H^{1/2}({\it{\Gamma}}) \setminus \{0\}$$, this proves (2.10). □ 2.6 Proof of Theorem 2.1 First, we prove boundedness and coercivity of $$a^\star(\cdot,\cdot)$$. Then, the Lions–Stampacchia theorem (see, e.g., Glowinski et al., 1981; Rodrigues, 1987; Glowinski, 2008) proves unique solvability of (2.4) provided that $$F : U\to {\mathbb{R}}$$ is a linear functional, which follows by the boundedness of $${\it{\Theta}}:U\to{\mathbb{R}}$$. We show the boundedness. Note that $$b:U\times V\to {\mathbb{R}}$$ and $${\it{\Theta}}:U\to V$$ are uniformly bounded, \begin{align*}|b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}})| \leq \beta C_2 \|{{\boldsymbol{u}}}\|_{U}\|{{\it{\Theta}}{\boldsymbol{v}}}\|_{V} \leq \beta C_2^2\|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U}\quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}}\in U. \end{align*} By duality and boundedness of $$\gamma_0,\gamma_{\boldsymbol{n}}$$ (see Lemma 2.7) we have \begin{align*}|\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq \|{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^{-1/2}({\it{\Gamma}})}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})} \leq \|{{\boldsymbol{u}}}\|_{U}\|{{\boldsymbol{v}}}\|_{U} \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}}\in U. \end{align*} Next, we follow Führer et al. (2017a) to prove coercivity. Note that $$a^0({\boldsymbol{u}},{\boldsymbol{u}}) = a^{\boldsymbol{n}}({\boldsymbol{u}},{\boldsymbol{u}}) = a^s({\boldsymbol{u}},{\boldsymbol{u}})$$ for all $${\boldsymbol{u}}\in U$$. Let $${\boldsymbol{u}}\in U$$. By the triangle inequality and Lemma 2.9 we get \begin{align*}\|{\boldsymbol{u}}\|_{U}^2 \lesssim \|{{\boldsymbol{u}}-{\mathcal{E}}\gamma_0{\boldsymbol{u}}}\|_U^2 + \|{{\mathcal{E}}\gamma_0{\boldsymbol{u}}}\|_{U}^2 \lesssim \|{B{\boldsymbol{u}}}\|_{V'}^2 + \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Let $$\widetilde u\in H^1({\it{\Omega}})$$ be the quasi-harmonic extension (2.8) of $$\gamma_0{\boldsymbol{u}}$$. The definition of the $$H^{1/2}({\it{\Gamma}})$$-norm and integration by parts show that \begin{align*}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 &= \|{\widetilde u}\|_{H^1({\it{\Omega}})}^2 = \|{\widetilde u}\|_{L^2({\it{\Omega}})}^2 + \|{\nabla \widetilde u}\|_{L^2({\it{\Omega}})}^2 = \langle{\partial_{\boldsymbol{n}} \widetilde u} ,{\widetilde u|_{\it{\Gamma}}}\rangle_{\it{\Gamma}} = \langle{\gamma_{\boldsymbol{n}} \mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} This gives \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} Using duality, Lemma 2.11 and Young’s inequality, we obtain \begin{align*}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} &\leq \|{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}\|_{H^{-1/2}({\it{\Gamma}})} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}\leq \sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \\ &\leq \|{B{\boldsymbol{u}}}\|_{V'}^2 + \frac{1}2 \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Altogether, this shows \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \tfrac12 \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \leq 2 \|{B{\boldsymbol{u}}}\|_{V'}^2 + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}. \end{align*} Thus, \begin{align*}C_1^{-1} \|{{\boldsymbol{u}}}\|_{U^2} \leq \beta\|{B{\boldsymbol{u}}}\|_{V'}^2 +\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}}{ ,\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} = a^\star({\boldsymbol{u}},{\boldsymbol{u}}) \end{align*} for all $$\beta\geq 2$$, where the constant $$C_1>0$$ depends only on $${\it{\Omega}}$$. Regarding the equivalence of problems (1.1) and (2.4) we know that the unique solution $$u$$ of (1.1) with $${\boldsymbol{u}}$$ defined as in the assertion satisfies (2.4) by construction. The other direction follows by existence of a unique solution of (2.4). Finally, note that (2.5) follows by construction. However, in the case $$\star=0$$ we can infer this identity directly from (2.4) as follows. Let $${\boldsymbol{u}}$$ denote the solution of (2.4) with $$\star=0$$. Set $${\boldsymbol{v}} = {\boldsymbol{u}}\pm ({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})$$ for some arbitrary $${\boldsymbol{w}}\in U$$. Since $$\gamma_0({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) = 0$$, we infer $${\boldsymbol{v}}\in K^0$$, so that we can use it as a test function in (2.4). This gives \begin{align*}\pm a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) \geq \pm F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}). \end{align*} Hence, \begin{align*}a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) = F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) \quad\text{for all }{\boldsymbol{w}}\in U. \end{align*} Note that $$\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}\in\ker B = \ker {\it{\Theta}}$$. This leads to \begin{align*}F({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) &= L({\it{\Theta}}_\beta ({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})) = L({\it{\Theta}}_\beta{\boldsymbol{w}}) = F({\boldsymbol{w}}) \text{ and }\\ a^0({\boldsymbol{u}},{\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}}) &= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{w}}) + \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{w}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{w}})}\rangle_{\it{\Gamma}}= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{w}}). \end{align*} This concludes the proof. 3. DPG method for a singularly perturbed problem In this section, we introduce and analyse a DPG method for the Signorini problem of reaction-dominated diffusion, i.e., for (1.1) with small positive constant $$c=\varepsilon$$ ($$0<\varepsilon\le 1$$). We stick mainly to the notation as given in Section 2 but need some additional definitions. Also, we redefine some objects like the bilinear form $$b:\;U\times V\to{\mathbb{R}}$$, the spaces $$U$$, $$V$$ and some norms. In fact, our objective of robust control of field variables forces us to carefully scale parts of norms with coefficients depending on the diffusion parameter $$\varepsilon$$. The ultra-weak formulation taken from Heuer & Karkulik (2017) is derived by rewriting problem (1.1) (with $$c=\varepsilon$$) as a first-order system \begin{align*}\rho-{\mathrm{div}}\,{\boldsymbol\sigma} = 0, \qquad \varepsilon^{-1/4}{\boldsymbol\sigma} - \nabla u = 0, \qquad -\varepsilon^{3/4}\rho + u = f. \end{align*} We test the first two equations with $$\mu\in H^1({\mathcal{T}})$$ resp. $${\boldsymbol\tau}\in {\boldsymbol{H}}({\mathrm{div}},{\mathcal{T}})$$ elementwise, integrate by parts and sum over all elements. The third equation is tested with $$v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v$$ for $$v\in H^1({\it{\Delta}},{\mathcal{T}})$$. Integrating by parts and using the first two equations, we get the formulation \begin{align}({\rho} ,{\mu}) + ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}}\mu}) - \langle{\widehat \sigma^a} ,{\mu}\rangle_{\mathcal{S}} &= 0, \\ \end{align} (3.1a) \begin{align}\varepsilon^{-1/4}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u^a} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= 0, \\ \end{align} (3.1b) \begin{align}\begin{split}\varepsilon^{3/4}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{3/4} \langle{\widehat\sigma^b} ,{v}\rangle_{\mathcal{S}} + ({u} ,{v}) \qquad\qquad\qquad\qquad& \quad\\ +{}\varepsilon^{5/4} ({\rho} ,{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) + \varepsilon^{1/4} ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{1/2} \langle{\widehat u^b} ,{\nabla_{\mathcal{T}}v\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} &= ({f} ,{v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v}). \end{split}\end{align} (3.1c) 3.1 Notation For functions $$\widehat u \in H^{1/2}({\mathcal{S}})$$, $$\widehat\sigma\in H^{-1/2}({\mathcal{S}})$$ we define the skeleton norms \begin{align*}\|{\widehat u}\|_{1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \varepsilon^{1/2}\|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), \widehat u|_{\partial{T}}=w|_{\partial{T}}\; \forall\,{T}\in{\mathcal{T}} \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\mathcal{S}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\varepsilon\|{{\mathrm{div}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}}), \widehat\sigma|_{\partial{T}}\!=\!({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{T}})|_{\partial{T}}\; \forall\,{T}\!\in\!{\mathcal{T}} \right\}. \end{align*} Moreover, for $$\widehat u\in H^{1/2}({\it{\Gamma}})$$, $$\widehat\sigma\in H^{1/2}({\it{\Gamma}})$$ we define the boundary norms \begin{align*}\|{\widehat u}\|_{1/2,{\it{\Gamma}}} &:= \inf \left\{ (\|{w}\|_{}^2 + \varepsilon^{1/2}\|{\nabla w}\|_{}^2)^{1/2} \,:\, w\in H^1({\it{\Omega}}), w|_{\it{\Gamma}}=\widehat u \right\},\\ \|{\widehat\sigma}\|_{-1/2,{\it{\Gamma}}} &:= \inf \left\{ (\|{{\boldsymbol{q}}}\|_{}^2\!+\!\varepsilon\|{{\mathrm{div}}{\boldsymbol{q}}}\|_{}^2)^{1/2} \,:\, {\boldsymbol{q}}\!\in\!{\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}}), ({\boldsymbol{q}}\cdot{\boldsymbol{n}}_{{\it{\Omega}}})|_{{\it{\Gamma}}}=\widehat\sigma \right\}. \end{align*} We will need another norm in $$H^{1/2}({\it{\Gamma}})$$ defined by \begin{align*}\|{\widehat u}\|_{H^{1/2}({\it{\Gamma}})}:= \inf \left\{{ \big(\|{w}\|_{}^2 + \varepsilon \|{\nabla w}\|_{}^2\big)^{1/2}} \,:\, {w \in H^1({\it{\Omega}}), w|_{\it{\Gamma}} = \widehat u} \right\}. \end{align*} Obviously, the latter norm is weaker than the previously defined corresponding boundary norm, $$\|\,{\cdot}\,\|_{H^{1/2}({\it{\Gamma}})}\le \|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}$$. The ultra-weak formulation from Heuer & Karkulik (2017) is based on the spaces $$U$$ and $$V$$ defined by \begin{align*}\widetilde U &:= L^2({\it{\Omega}}) \times [L^2({\it{\Omega}})]^d \times L^2({\it{\Omega}}) \times H^{1/2}({\mathcal{S}}) \times H^{1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}) \times H^{-1/2}({\mathcal{S}}), \\ U &:= \left\{{(u,{\boldsymbol\sigma},\rho,\widehat u^a, \widehat u^b, \widehat\sigma^a,\widehat\sigma^b) \in \widetilde U} \,:\, {\widehat u^a|_{\it{\Gamma}} = \widehat u^b|_{\it{\Gamma}}} \right\}, \\ V &:= H^1({\mathcal{T}}) \times {\boldsymbol{H}}({\mathrm{div}},{\mathcal{T}}) \times H^1({\it{\Delta}},{\mathcal{T}}), \quad\text{where}\\ H^1({\it{\Delta}},{\mathcal{T}}) &:= \{w\in H^1({\mathcal{T}}) \,:\, {\it{\Delta}} w|_T \in L^2(T) \,\forall\, T\in{\mathcal{T}}\}. \end{align*} For the analysis we need two different norms in $$U$$: \begin{align*}\|{{\boldsymbol{u}}}\|_{U,1}^2 &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \\ &\quad + \varepsilon^{3/2}\|{\widehat u^a}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon^{3/2}\|{\widehat \sigma^a}\|_{-1/2,{\mathcal{S}}}^2 + \varepsilon^{5/2}\|{\widehat \sigma^b}\|_{-1/2,{\mathcal{S}}}^2, \\ \|{{\boldsymbol{u}}}\|_{U,2}^2 &:= \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \\ &\quad + \|{\widehat u^a}\|_{1/2,{\mathcal{S}}}^2 + \varepsilon^{-1/2} \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 + \|{\widehat \sigma^a}\|_{-1/2,{\mathcal{S}}}^2 + \varepsilon^{1/2}\|{\widehat \sigma^b}\|_{-1/2,{\mathcal{S}}}^2 \end{align*} for $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in U$$. These norms differ in their $$\varepsilon$$-scalings of the skeleton components so that $$\|\,{\cdot}\,\|_{U,1}\le \|\,{\cdot}\,\|_{U,2}$$. In both cases, field components are measured in the so-called balanced norm$$(\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2)^{1/2}$$ which, for the exact solution, is $$(\|{u}\|_{}^2 + \varepsilon^{1/2}\|{\nabla u}\|_{}^2 + \varepsilon^{3/2} \|{{\it{\Delta}} u}\|_{}^2)^{1/2}$$ (cf. Lin & Stynes, 2012; Heuer & Karkulik, 2017). The test space $$V$$ is equipped with the norm \begin{align*}\|{{\boldsymbol{v}}}\|_{V}^2 &:= \varepsilon^{-1}\|{\mu}\|_{}^2 + \|{\nabla_{\mathcal{T}}\mu}\|_{}^2 + \varepsilon^{-1/2} \|{{\boldsymbol\tau}}\|_{}^2 + \|{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}\|_{}^2 \\ &\quad + \|{v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla_{\mathcal{T}} v}\|_{}^2 + \varepsilon^{3/2}\|{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}\|_{}^2 \quad\text{for }{\boldsymbol{v}} = (\mu,{\boldsymbol\tau},v)\in V. \end{align*} This norm is induced by the inner product $$(\cdot ,\cdot)_V$$ on $$V$$. Note that this norm is equivalent to the one defined in Heuer & Karkulik (2017). The only difference is that the term $$\varepsilon\|{\nabla_{\mathcal{T}} v}\|_{}^2$$ is not present in Heuer & Karkulik (2017). We use this norm here to get a smaller constant in Lemma 3.10 below. 3.2 Ultra-weak formulation The left- and right-hand sides of (3.1) give rise to the following definitions of the bilinear form $$b : U\times V \to {\mathbb{R}}$$ and the linear functional $$L : V \to {\mathbb{R}}$$: \begin{align*}b({\boldsymbol{u}},{\boldsymbol{v}}) &:= ({\rho} ,{\mu}) + ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}}\mu}) - \langle{\widehat \sigma^a} ,{\mu}\rangle_{\mathcal{S}} + \varepsilon^{-1/4}({{\boldsymbol\sigma}} ,{{\boldsymbol\tau}}) + ({u} ,{{\rm div}_{{\mathcal{T}}}\,{\boldsymbol\tau}}) - \langle{\widehat u^a} ,{{\boldsymbol\tau}\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}} \\ &\quad +\varepsilon^{3/4}({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{3/4} \langle{\widehat\sigma^b} ,{v}\rangle_{\mathcal{S}} + ({u} ,{v}) \\ &\quad +\varepsilon^{5/4} ({\rho} ,{{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) + \varepsilon^{1/4} ({{\boldsymbol\sigma}} ,{\nabla_{\mathcal{T}} v}) - \varepsilon^{1/2} \langle{\widehat u^b} ,{\nabla_{\mathcal{T}} v\cdot{\boldsymbol{n}}}\rangle_{\mathcal{S}},\\ L({\boldsymbol{v}}) &:= ({f} ,{v-\varepsilon^{1/2}{{\it{\Delta}}}_{{\mathcal{T}}}\, v}) \end{align*} for all $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b, \widehat\sigma^a,\widehat\sigma^b) \in U$$, $${\boldsymbol{v}} = (\mu,{\boldsymbol\tau},v) \in V$$. The trial-to-test operator $${\it{\Theta}}_\beta : U \to V$$ is defined as before; see (1.5). Again, the operator $$B:\;U \to V'$$ is induced by the bilinear form $$b(\cdot,\cdot)$$ and (3.1) can be written as \begin{align*}\text{find }{\boldsymbol{u}}\in U \text{ s.t.}\quad B{\boldsymbol{u}} = L. \end{align*} The nontrivial kernel of $$B$$ is related to the trace operators. For the present space $$U$$ we define them by \begin{alignat*}{2}\gamma_0 &: U \to H^{1/2}({\it{\Gamma}}), &\qquad \gamma_0{\boldsymbol{u}} &:= \widehat u^a|_{\it{\Gamma}}, \\ \gamma_{\boldsymbol{n}} &: U \to H^{-1/2}({\it{\Gamma}}), &\qquad \gamma_{\boldsymbol{n}}{\boldsymbol{u}} &:= \widehat\sigma^a|_{\it{\Gamma}}. \end{alignat*} For simplicity, we consider only the symmetric formulation. The other cases can be derived similarly; see also Section 2. Analogously to the unperturbed case we introduce the nonempty convex subset \begin{align*}K^s = \left\{{ {\boldsymbol{u}}\in U} \,:\, {\gamma_0{\boldsymbol{u}}\geq 0, \gamma_{\boldsymbol{n}}{\boldsymbol{u}} \geq 0} \right\}\end{align*} and define the bilinear form $$a^s:\;U\times U\to{\mathbb{R}}$$ and linear functional $$F:\;U\to{\mathbb{R}}$$ by \begin{align*}a^s({\boldsymbol{u}},{\boldsymbol{v}}) &:= b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}}) + \tfrac12 \varepsilon^{1/4} ( \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}+\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} ), \\ F({\boldsymbol{v}}) &:= L({\it{\Theta}}_\beta{\boldsymbol{v}}) \end{align*} for $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. Here, $$\beta>0$$ is a constant to be fixed. Then, our variational inequality reads, find $${\boldsymbol{u}}\in K^s$$ such that \begin{align}\label{eq:sp:varineq}a^s({\boldsymbol{u}},{\boldsymbol{v}}-{\boldsymbol{u}}) \geq F({\boldsymbol{v}}-{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{v}}\in K^s. \end{align} (3.2) In the singularly perturbed case it is convenient to state coercivity and boundedness of the bilinear form $$a^s(\cdot,\cdot)$$ in the energy-based norm \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 := \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \quad ({\boldsymbol{u}}\in U). \end{align*} Note that $$\|{B\,\cdot}\|{V'}$$ is the energy norm in standard DPG settings, whereas in our case it is a seminorm. Corresponding to Theorem 2.1 we have the following result. Theorem 3.1 For all $$\beta\geq 3$$ the bilinear form $$a^s:U\times U \to {\mathbb{R}}$$ is coercive, \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 \leq C_1 a^s({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all } {\boldsymbol{u}}\in U, \end{align*} and bounded, \begin{align*}|a^s({\boldsymbol{u}},{\boldsymbol{v}})| \leq C_2 | | |{{\boldsymbol{u}}}| | || | |{{\boldsymbol{v}}}| | | \quad\text{for all }{\boldsymbol{u}},{\boldsymbol{v}} \in U. \end{align*} The constants $$C_1,C_2>0$$ do not depend on $${\it{\Omega}}$$, $${\mathcal{T}}$$ or $$\varepsilon$$. Also $$C_1$$ is independent of $$\beta$$ but $$C_2$$ is not. Furthermore, \begin{align}\label{eq:sp:normequiv}\|{{\boldsymbol{u}}}\|_{U,1} \lesssim | | |{{\boldsymbol{u}}}| | | \lesssim \|{{\boldsymbol{u}}}\|_{U,2} \quad\text{for all } {\boldsymbol{u}}\in U, \end{align} (3.3) with generic constants that are independent of $${\mathcal{T}}$$ and $$\varepsilon$$. The variational inequality (3.2) is uniquely solvable and equivalent to problem (1.1) (setting $$c=\varepsilon$$) in the following sense. If $$u\in H^1({\it{\Omega}})$$ solves problem (1.1), then $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in K^s$$ with $${\boldsymbol\sigma} := \varepsilon^{1/4}\nabla u$$, $$\widehat u^\star|_{\partial T} := u|_{\partial T}$$, $$\widehat\sigma^\star|_{\partial T} := \sigma\cdot{\boldsymbol{n}}_T|_{\partial T}$$ for all $$T\in{\mathcal{T}}$$ ($$\star\in\{a,b\}$$) solves (3.2). On the other hand, if $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)\in K^s$$ solves (3.2), then $$u\in H^1({\it{\Omega}})$$ solves (1.1). Moreover, the unique solution $${\boldsymbol{u}}\in K^s$$ of (3.2) satisfies \begin{align*}b({\boldsymbol{u}},{\it{\Theta}}_\beta {\boldsymbol{w}}) = F({\boldsymbol{w}}) \quad\text{for all } {\boldsymbol{w}}\in U. \end{align*} We prove this result in Section 3.5. 3.3 Discretization, convergence and a posteriori error estimate We replace $$U$$ by the lowest-order subspace \begin{align*}U_h := P^0({\mathcal{T}}) \times [P^0({\mathcal{T}})]^d \times P^0({\mathcal{T}}) \times S^1({\mathcal{S}}) \times S^1({\mathcal{S}}) \times P^0({\mathcal{S}}) \times P^0({\mathcal{S}}) \end{align*} and $$K^s$$ by \begin{align*}K_h^s := \left\{{{\boldsymbol{v}}_h\in U_h} \,:\, {\gamma_0{\boldsymbol{v}}_h\geq 0,\gamma_{\boldsymbol{n}}{\boldsymbol{v}}_h\geq 0} \right\}. \end{align*} The discrete version of (3.2) then reads, find $${\boldsymbol{u}}_h\in K_h^s$$ such that \begin{align}\label{eq:sp:varineqdisc}a^s({\boldsymbol{u}}_h,{\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \geq F({\boldsymbol{v}}_h-{\boldsymbol{u}}_h) \quad\text{for all } {\boldsymbol{v}}_h \in K_h^s. \end{align} (3.4) With the same arguments as in Section 2.3 we can prove unique solvability. Theorem 3.2 Under the same assumptions as in Theorem 3.1, the discrete variational inequality (3.4) admits a unique solution $${\boldsymbol{u}}_h\in K_h^s$$. We have the following robust quasi-optimal a priori error estimate. Here, robustness means that the hidden constant does not depend on the positive perturbation parameter $$\varepsilon$$ (though, for simplicity, we have assumed that $$\varepsilon\le 1$$). Theorem 3.3 For $$\beta\ge 3$$ let $${\boldsymbol{u}}\in K^s$$, $${\boldsymbol{u}}_h\in K_h^s$$ denote the exact solutions of (3.2), (3.4). Then, it holds that \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U,1}^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( \|{{\boldsymbol{u}}-{\boldsymbol{v}}_h}\|_{U,2}^2 + \tfrac12\varepsilon^{1/4}(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right)\!. \end{align*} The generic constant does depend on $${\it{\Omega}}$$ and $$\beta$$ but not on $${\mathcal{T}}$$ or $$\varepsilon$$. Proof. Following the proof of Theorem 2.3 we obtain (by replacing $$\|\,{\cdot}\,\|_U$$ with $$| | |\cdot| | |$$) \begin{align*}| | |{{\boldsymbol{u}}-{\boldsymbol{u}}_h}| | |^2 \lesssim \inf_{{\boldsymbol{v}}_h\in K_h^s} \left( | | |{{\boldsymbol{u}}-{\boldsymbol{v}}_h}| | |^2 + \tfrac12\varepsilon^{1/4}(\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0({\boldsymbol{v}}_h-{\boldsymbol{u}})}\rangle_{\it{\Gamma}}+ \langle{\gamma_{\boldsymbol{n}}({\boldsymbol{v}}_h-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}})\right)\!. \end{align*} Then, (3.3) proves the error bound. □ The derivation of an a posteriori error estimate is analogous to Section 2.4. For a function $${\boldsymbol{u}}_h\in K_h^s$$ we define local error indicators by \begin{align*}\eta(T)^2 &:= \beta \|{R_T^{-1} \iota_T^*(L-B{\boldsymbol{u}}_h)}\|_{V(T)}^2, \\ \eta(E)^2 &:= \varepsilon^{1/4} \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}_h} ,{\gamma_0{\boldsymbol{u}}_h}\rangle_E, \end{align*} and the overall estimator \begin{align*}\eta^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2 + \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} Here, $$V(T) := H^1(T) \times {\boldsymbol{H}}({\mathrm{div}},T) \times H^1({\it{\Delta}},T)$$ is equipped with the norm \begin{align*}\|{(\mu,{\boldsymbol\tau},v)}\|_{V(T)}^2 &:= \varepsilon^{-1}\|{\mu}\|_{T}^2 + \|{\nabla\mu}\|_{T}^2 + \varepsilon^{-1/2} \|{{\boldsymbol\tau}}\|_{T}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{T}^2 \\ &\quad{} + \|{v}\|_{T}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{T}^2 + \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{T}^2, \end{align*} where $$R_T:V(T) \to (V(T))'$$ denotes the Riesz isomorphism and $$\iota_T^*$$ is the dual of the canonical embedding $$\iota_T:V(T)\to V$$. Analogous to the proof of Theorem 2.6, in conjunction with (3.3), we obtain the following a posteriori estimate. Like the a priori estimate from Theorem 3.3, the a posteriori estimate is robust with respect to $$\varepsilon$$ ($$0<\varepsilon\le 1$$). Theorem 3.4 For $$\beta\ge 3$$ let $${\boldsymbol{u}}\in K^s$$ and $${\boldsymbol{u}}_h\in K_h^s$$ be the solutions of (3.2) and (3.4), respectively. Then, there holds the reliability estimate \begin{align*}| | |{{\boldsymbol{u}}-{\boldsymbol{u}}_h}| | | \leq C_\mathrm{rel} \eta. \end{align*} The constant $$C_\mathrm{rel}>0$$ is independent of $${\it{\Omega}}$$, $${\mathcal{T}}$$, $$\beta$$ and $$\varepsilon$$. In particular, with (3.3) we have \begin{align*}\|{{\boldsymbol{u}}-{\boldsymbol{u}}_h}\|_{U,1} \leq C_{\mathrm{rel},U} \eta, \end{align*} where $$C_{\mathrm{rel},U} := C_\mathrm{rel} C_1$$ and $$C_1$$ depends only on $${\it{\Omega}}$$. 3.4 Technical details Analogously to Lemma 2.7, we obtain boundedness of the trace operators. In this case, we use that, by definition of the norms, $$\|{\widehat u^a|_{\it{\Gamma}}}\|_{1/2,{\it{\Gamma}}}\leq \|{\widehat u^a}\|_{1/2,{\mathcal{S}}}$$ and $$\|{\widehat\sigma^a|_{\it{\Gamma}}}\|_{-1/2,{\it{\Gamma}}}\leq \|{\widehat\sigma^a}\|_{-1/2,{\mathcal{S}}}$$. Lemma 3.5 The operators $$\gamma_0 : \big(U,\|\,{\cdot}\,\|_{U,2}\big) \to \big(H^{1/2}({\it{\Gamma}}),\|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}\big)$$ and $$\gamma_{\boldsymbol{n}} : \big(U,\|\,{\cdot}\,\|_{U,2}\big) \to \big( H^{-1/2}({\it{\Gamma}}), \|\,{\cdot}\,\|_{-1/2,{\it{\Gamma}}}\big)$$ have unit norm. We now adapt the definition of the previously employed extension operator $$\mathcal{E}$$ to the current situation. For a function $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ we define its quasi-harmonic extension $$\widetilde u \in H^1({\it{\Omega}})$$ as the unique solution of \begin{align}\label{eq:sp:harmext}-\varepsilon {\it{\Delta}} \widetilde u + \widetilde u &=0 \quad\text{in }{\it{\Omega}}, \qquad \widetilde u|_{\it{\Gamma}} = \widehat v, \end{align} (3.5) and define $$\mathcal{E} : H^{1/2}({\it{\Gamma}}) \to U$$ by \begin{align*}\mathcal{E} \widehat v := \left(\widetilde u, \varepsilon^{1/4}\nabla \widetilde u, \varepsilon^{1/4} {\it{\Delta}} \widetilde u, \widetilde u|_{{\mathcal{S}}}, \widetilde u|_{{\mathcal{S}}}, \varepsilon^{1/4} \left(\nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T}\right)_{T\in{\mathcal{T}}}, \varepsilon^{1/4} \left(\nabla\widetilde u\cdot{\boldsymbol{n}}_T|_{\partial T}\right)_{T\in{\mathcal{T}}}\right). \end{align*} This operator characterizes the kernel of $$B$$. Note that, in contrast to $$\|\,{\cdot}\,\|_{1/2,{\it{\Gamma}}}$$, the norm $$\|\,{\cdot}\,\|_{H^{1/2}({\it{\Gamma}})}$$ (recall the definitions in Section 3.1) is inherited from the energy norm associated with problem (3.5). Lemma 3.6 For given $$\widehat v\in H^{1/2}({\it{\Gamma}}),$$ let $$\widetilde u \in H^1({\it{\Omega}})$$ be the unique solution of (3.5). Then, \begin{align*}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2 = \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 = \varepsilon \langle{\partial_{\boldsymbol{n}} \widetilde u} ,{\widehat v}\rangle_{\it{\Gamma}} = \varepsilon^{3/4}\langle{\gamma_{\boldsymbol{n}} \mathcal{E} \widehat v} ,{\widehat v}\rangle_{\it{\Gamma}}. \end{align*} Proof. The last identity follows by definition of the operator $$\mathcal{E}$$. Using the weak formulation of problem (3.5) we have \begin{align*}\varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 &= \varepsilon\langle{\partial_{\boldsymbol{n}}\widetilde u} ,{\widehat v}\rangle_{\it{\Gamma}}= \varepsilon({\nabla \widetilde u} ,{\nabla w}) + ({\widetilde u} ,{w}) \\ &\leq \Big( \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2\Big)^{1/2}\Big( \varepsilon\|{\nabla w}\|_{}^2 + \|{w}\|_{}^2\Big)^{1/2}\end{align*} for all $$w\in H^1({\it{\Omega}})$$ with $$w|_{\it{\Gamma}} = \widehat v$$. Thus, $$\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2 = \varepsilon\|{\nabla \widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2$$. □ Similarly to Lemma 2.9, there holds the following result in the singularly perturbed case. Lemma 3.7 The operators $$B : U \to V'$$, $$\mathcal{E}: H^{1/2}({\it{\Gamma}})\to U$$ have the following properties: (i) The operators $$B$$ and $$\mathcal{E}$$ are bounded: \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'} &\lesssim \|{{\boldsymbol{u}}}\|_{U,2} \quad\text{for all }{\boldsymbol{u}}\in U, \qquad \|{\mathcal{E} \widehat v}\|_{U,1} \simeq \varepsilon^{-1/4} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}\quad\text{for all } \widehat v \in H^{1/2}({\it{\Gamma}}). \end{align*} The generic constants are independent of $${\mathcal{T}}$$ and $$\varepsilon$$. (ii) The kernel of $$B$$ consists of all quasi-harmonic extensions, i.e., $$\ker(B) = \operatorname{ran}(\mathcal{E})$$. (iii) The operator $$\mathcal{E}$$ is a right inverse of $$\gamma_0$$. (iv) The operator $$B:\;U/\ker(B)\to V'$$ is inf–sup stable: \begin{align*}\|{{\boldsymbol{\it{u}}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1} \lesssim \|{B{\boldsymbol{u}}}\|_{V'} = \sup_{{\boldsymbol{\it{v}}}\in V} \frac{\langle B\mathbf{u},{\boldsymbol{v}}\rangle }{\|{\boldsymbol{v}}{{\|}_{V}}}= b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} \quad\text{for all } {\boldsymbol{u}}\in U. \end{align*} The generic constant depends on $${\it{\Omega}}$$ but not on $${\mathcal{T}}$$ or $$\varepsilon$$. Proof. First, by Heuer & Karkulik (2017, Lemma 3), we have $$b({\boldsymbol{u}},{\boldsymbol{v}})\lesssim\|{{\boldsymbol{u}}}\|_{U,2}\|{{\boldsymbol{v}}}\|_{V}$$ for all $${\boldsymbol{u}}\in U,{\boldsymbol{v}}\in V$$. Dividing by $$\|{{\boldsymbol{v}}}\|_{V}$$ and taking the supremum proves boundedness of $$B$$. Second, for $$\widehat v\in H^{1/2}({\it{\Gamma}})$$, let $${\boldsymbol{u}} = (u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b) = \mathcal{E} \widehat v$$. By definition of the skeleton norms we have \begin{align*}&\|{\widehat u^\star}\|_{1/2,{\mathcal{S}}}^2 \leq \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2,\quad\|{\widehat \sigma^\star}\|_{-1/2,{\mathcal{S}}}^2 \leq \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon\|{\rho}\|_{}^2 \quad (\star\in\{a,b\}). \end{align*} In the definition of $$\|\,{\cdot}\,\|_{U,1}$$, these norms are scaled with positive powers of $$\varepsilon$$. Thus, \begin{align*}\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2 \leq \|{{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon \|{\rho}\|_{}^2. \end{align*} Using that $$\rho = \varepsilon^{1/4} {\it{\Delta}} u = \varepsilon^{1/4} \varepsilon^{-1} u = \varepsilon^{-3/4} u$$ and $${\boldsymbol\sigma} = \varepsilon^{1/4}\nabla u$$ by the definition of $${\boldsymbol{u}}=\mathcal{E}\widehat v$$ (cf. (3.5)) we obtain \begin{align*}\|{u}\|_{}^2 + \|{{\boldsymbol\sigma}}\|_{}^2 + \varepsilon\|{\rho}\|_{}^2 = \|{u}\|_{}^2 + \varepsilon^{1/2} \|{\nabla u}\|_{}^2 + \varepsilon^{-1/2}\|{u}\|_{}^2 \simeq \varepsilon^{-1/2}\left( \|{u}\|_{}^2 + \varepsilon \|{\nabla u}\|_{}^2 \right). \end{align*} The last term on the right-hand side is equal to $$\varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2$$ by Lemma 3.6. Finally, (ii), (iii) and (iv) are proved in Führer & Heuer (2016, Lemmas 3, 4). □ Similarly to Lemma 2.11 (for the unperturbed case), we need to control the Neumann traces of elements of the quotient space $$U/\ker(B)$$. As we have seen, this has a fundamental relation to the stability of the homogeneous adjoint problem with prescribed Dirichlet boundary condition; cf. Lemma 2.10. In the singularly perturbed case, the situation is a little more technical. Following Heuer & Karkulik (2017) (see Lemmas 8 and 9 there), we split the stability analysis of the adjoint problem into two parts. These are the following Lemmas 3.8 and 3.9. The last lemma of this section (Lemma 3.10) then states the control of the Neumann traces. Lemma 3.8 Let $$\widehat w\in H^{1/2}({\it{\Gamma}})$$ be given. The problem \begin{align}{\mathrm{div}}{\boldsymbol\lambda} + \varepsilon^{-1/2}w &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualestimate:a} \\ \end{align} (3.6a) \begin{align}{\boldsymbol\lambda} + \nabla w &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualestimate:b} \\ \end{align} (3.6b) \begin{align}w|_{{\it{\Gamma}}} &= \widehat w \label{eq:sp:dualestimate:c}\end{align} (3.6c) admits a unique solution $$(w,{\boldsymbol\lambda}) \in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ with $${\it{\Delta}} w\in H^1({\it{\Omega}})$$, and \begin{align*}\|{{\boldsymbol\lambda}}\|_{}^2 + \varepsilon^{1/2}\|{{\mathrm{div}}{\boldsymbol\lambda}}\|_{}^2 = \|{\nabla w}\|_{}^2 + \varepsilon^{-1/2}\|{w}\|_{}^2 \leq \varepsilon^{-1} \|{\widehat w}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} Proof. The proof follows the same arguments as given in the proof of Lemma 2.10. To see the estimate for the norms, we make use of the weak formulation and obtain \begin{align*}\|{\nabla w}\|_{}^2 + \varepsilon^{-1/2}\|{w}\|_{}^2 \leq \|{\nabla \widetilde u}\|_{}^2 + \varepsilon^{-1/2}\|{\widetilde u}\|_{}^2 \leq \varepsilon^{-1} \left( \varepsilon\|{\nabla\widetilde u}\|_{}^2 + \|{\widetilde u}\|_{}^2 \right) \end{align*} for all $$\widetilde u\in H^1({\it{\Omega}})$$ with $$\widetilde u|_{\it{\Gamma}} = \widehat w$$. Taking the infimum over these functions $$\widetilde u$$ finishes the proof. □ Lemma 3.9 Let $$\widehat v\in H^{1/2}({\it{\Gamma}})$$ be given. The problem \begin{align}{\mathrm{div}}{\boldsymbol\tau} + v &= 0\quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:a} \\ \end{align} (3.7a) \begin{align}\nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau} &= 0 \quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:b}\\ \end{align} (3.7b) \begin{align}\varepsilon^{5/4} {\it{\Delta}} v + \mu &= 0\quad\text{in } {\it{\Omega}}, \label{eq:sp:dualproblem:c} \\ \end{align} (3.7c) \begin{align}\quad v|_{\it{\Gamma}} = 0, \qquad {\it{\Delta}} v|_{\it{\Gamma}} = -\varepsilon^{-5/4} {\widehat v}\end{align} (3.7d) has a unique solution $${\boldsymbol{v}}:=(\mu,{\boldsymbol\tau},v)\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})\times H^1({\it{\Delta}},{\it{\Omega}})$$. It satisfies $$\mu|_{\it{\Gamma}} = \widehat v$$ and \begin{align*}\|{{\boldsymbol{v}}_{V}}\| \leq 3/\sqrt{2} \varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} Proof. Let $$(w,{\boldsymbol\lambda})\in H^1({\it{\Omega}})\times {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ be the solution of (3.6) with $$\widehat w = \varepsilon^{-1/4}\widehat v$$. Define $$v\in H_0^1({\it{\Omega}})$$ to be the unique solution of \begin{align*}-\varepsilon {\it{\Delta}} v + v = w\quad\text{in } {\it{\Omega}}, \qquad v|_{\it{\Gamma}} = 0. \end{align*} Note that $${\it{\Delta}} v = \varepsilon^{-1} (v-w) \in H^1({\it{\Omega}})$$ and $${\it{\Delta}} v|_{\it{\Gamma}} = -\varepsilon^{-1}w|_{\it{\Gamma}} = -\varepsilon^{-5/4}\widehat v$$. We define $$\mu:= -\varepsilon^{5/4}{\it{\Delta}} v\in H^1({\it{\Omega}})$$ and have $$\mu|_{\it{\Gamma}} = \varepsilon^{1/4}w|_{\it{\Gamma}} = \widehat v$$. In particular, (3.7c) and (3.7d) are satisfied. Now, define $${\boldsymbol\tau} := \varepsilon^{1/2}{\boldsymbol\lambda} - \varepsilon\nabla v$$. Note that $${\boldsymbol\lambda}\in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$ and $$\nabla v \in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$. Hence, $${\boldsymbol\tau}\in {\boldsymbol{H}}({\mathrm{div}},{\it{\Omega}})$$. Together with (3.6a) and $$-\varepsilon{\it{\Delta}} v + v = w$$ we get \begin{align*}{\mathrm{div}}{\boldsymbol\tau} = \varepsilon^{1/2}{\mathrm{div}}{\boldsymbol\lambda} - \varepsilon{\it{\Delta}} v = -w - \varepsilon{\it{\Delta}} v = -v, \end{align*} which is (3.7a). One also establishes that (3.7b) holds. In fact, by the definition of $${\boldsymbol\tau}$$ and relation (3.6b), we find \begin{align*}\nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau}&= \nabla\mu + (\varepsilon^{1/4}+\varepsilon^{3/4})\nabla v + \varepsilon^{-1/4} (\varepsilon^{1/2}{\boldsymbol\lambda}-\varepsilon\nabla v)\\ &= \nabla\mu + \varepsilon^{1/4}\nabla v - \varepsilon^{1/4}\nabla w. \end{align*} The last term vanishes since $$\mu=-\varepsilon^{5/4}{\it{\Delta}} v = -\varepsilon^{1/4}(v-w)$$ by definition of $$\mu$$ and $$v$$. Now, testing (3.7c) with $$\varepsilon^{1/4}{\it{\Delta}} z$$, (3.7b) with $$\varepsilon^{1/4}\nabla z$$ and (3.7a) with $$z$$ for $$z\in H^1({\it{\Delta}},{\it{\Omega}})$$ with $$z|_{\it{\Gamma}} = 0$$, and adding the resulting equations, we obtain, after integrating by parts, \begin{align}\label{eq:sp:dualproblem:varform}\varepsilon^{3/2}({{\it{\Delta}} v} ,{{\it{\Delta}} z}) + (\varepsilon^{1/2}+\varepsilon) ({\nabla v} ,{\nabla z}) + ({v} ,{z}) = -\varepsilon^{1/4} \langle{\nabla z\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{\widehat v}\rangle_{\it{\Gamma}}. \end{align} (3.8) For any $$\widetilde u\in H^1({\it{\Omega}})$$ with $$\widetilde u|_{\it{\Gamma}} = \widehat v$$ we infer \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 &= -\varepsilon^{1/4}\langle{\nabla v\cdot{\boldsymbol{n}}_{\it{\Omega}}} ,{\widehat v}\rangle_{\it{\Gamma}}= -\varepsilon^{1/4} ({{\it{\Delta}} v} ,{\widetilde u}) -\varepsilon^{1/4}({\nabla v} ,{\nabla\widetilde u)} \\ &\leq \varepsilon^{3/4}\|{{\it{\Delta}} v}\|_{} \|{\varepsilon^{-1/2}\widetilde u}\|_{} + \varepsilon^{1/4}\|{\nabla v}\|_{}\|{\nabla \widetilde u}\|_{} \\ &\leq \left(\varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \varepsilon^{1/2}\|{\nabla v}\|_{}^2\right)^{1/2}\varepsilon^{-1/2}\left( \|{\widetilde u}\|_{}^2 + \varepsilon \|{\nabla \widetilde u}\|_{}^2\right)^{1/2}. \end{align*} On the one hand, we conclude \begin{align}\label{eq:Hm12estimate:estV1}\left(\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right)^{1/2}\leq \varepsilon^{-1/2}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}. \end{align} (3.9) On the other hand, using Young’s inequality, we also conclude that \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 \leq \tfrac{\delta^{-1}}2 \left(\varepsilon^{3/2} \|{{\it{\Delta}} v}\|_{}^2 + \varepsilon^{1/2}\|{\nabla v}\|_{}^2\right) + \tfrac{\delta}2 \varepsilon^{-1} \|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} For $$\delta=\tfrac12$$ we get \begin{align}\label{eq:Hm12estimate:estV2}\varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 \leq \tfrac14 \varepsilon^{-1}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align} (3.10) By (3.7c) and (3.7a) we have \begin{align*}\varepsilon^{-1}\|{\mu}\|_{}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{}^2 = \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \|{v}\|_{}^2. \end{align*} It remains to estimate the norms of $${\boldsymbol\tau}$$ and $$\nabla v$$. To this end, we rewrite the term $$({{\boldsymbol\lambda}} ,{\nabla v})$$. Integrating by parts, the condition $$v|_{\it{\Gamma}} = 0$$, (3.6a) and the identity $$w=-\varepsilon{\it{\Delta}} v + v$$ show that \begin{align*}({{\boldsymbol\lambda}} ,{\nabla v}) = -({{\mathrm{div}}{\boldsymbol\lambda}} ,{v}) &= \varepsilon^{-1/2}({w} ,{v}) = -\varepsilon^{1/2}({{\it{\Delta}} v} ,{v}) + \varepsilon^{-1/2}({v} ,{v}) = \varepsilon^{-1/2}\left( \varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right). \end{align*} Recall that $$\varepsilon^{-1/4}{\boldsymbol\tau} = \varepsilon^{1/4}{\boldsymbol\lambda} -\varepsilon^{3/4}\nabla v$$. Thus, \begin{align*}\varepsilon^{-1/2}\|{{\boldsymbol\tau}}\|_{}^2 = \varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2 -2\varepsilon({{\boldsymbol\lambda}} ,{\nabla v}) + \varepsilon^{3/2}\|{\nabla v}\|_{}^2. \end{align*} For the estimation of $$\|{\nabla \mu}\|_{}$$ we use (3.7b) and again $$\varepsilon^{1/4}{\boldsymbol\lambda} = \varepsilon^{3/4}\nabla v + \varepsilon^{-1/4}{\boldsymbol\tau}$$ to obtain \begin{align*}\|{\nabla\mu}\|_{}^2 = \|{\varepsilon^{1/4}\nabla v + \varepsilon^{1/4}{\boldsymbol\lambda}}\|_{}^2 = \varepsilon^{1/2}\|{\nabla v}\|_{}^2 + 2\varepsilon^{1/2}({{\boldsymbol\lambda}} ,{\nabla v}) + \varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2. \end{align*} This gives the estimate \begin{align*}&\varepsilon^{-1}\|{\mu}\|_{}^2 + \|{{\mathrm{div}}\,{\boldsymbol\tau}}\|_{}^2 + \varepsilon^{-1/2}\|{{\boldsymbol\tau}}\|_{}^2 + \|{\nabla\mu}\|_{}^2 \\ &\qquad \leq \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + \|{v}\|_{}^2 + 2\varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2 + (\varepsilon^{1/2}+2\varepsilon)\|{\nabla v}\|_{}^2 + 2\|{v}\|_{}^2 \\ &\qquad \leq \varepsilon^{3/2}\|{{\it{\Delta}} v}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2 + 2\left(\varepsilon\|{\nabla v}\|_{}^2 + \|{v}\|_{}^2\right) + 2\varepsilon^{1/2}\|{{\boldsymbol\lambda}}\|_{}^2. \end{align*} Using estimates (3.9), (3.10) and Lemma 3.8, we put everything together to conclude for the overall norm \begin{align*}\|{(\mu,{\boldsymbol\tau},v)}\|_{V}^2 \leq \frac{9}2 \varepsilon^{-1}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} To see uniqueness of $${\boldsymbol{v}}$$, let $$(\mu_2,{\boldsymbol\tau}_2,v_2)$$ solve (3.7). Define $${\boldsymbol{w}}:={\boldsymbol{v}}-(\mu_2,{\boldsymbol\tau}_2,v_2)$$ and set $$w:=v-v_2$$. Note that $$w|_{\it{\Gamma}} = 0$$ and $${\it{\Delta}} w|_{\it{\Gamma}} = 0$$. The variational formulation for $$w$$, which can be obtained in the same way as (3.8), proves \begin{align*}\varepsilon^{3/2} \|{{\it{\Delta}} w}\|_{}^2 + (\varepsilon^{1/2}+\varepsilon)\|{\nabla w}\|_{}^2 + \|{w}\|_{}^2 = 0. \end{align*} Thus, $$w=0$$ or equivalently $$v=v_2$$. Equation (3.7c) then gives $$\mu = -\varepsilon^{5/4}{\it{\Delta}} v = -\varepsilon^{5/4}{\it{\Delta}} v_2 = \mu_2$$ and, similarly, (3.7b) shows $${\boldsymbol\tau}={\boldsymbol\tau}_2$$. □ Lemma 3.10 It holds that \begin{align*}\varepsilon^{1/4}|\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\widehat v}\rangle_{\it{\Gamma}}| \leq 3/\sqrt{2}\|{B{\boldsymbol{u}}}\|_{V'}\varepsilon^{-1/4}\|{\widehat v}\|_{H^{1/2}({\it{\Gamma}})} \quad\text{for all } {\boldsymbol{u}}\in U, \, \widehat v\in H^{1/2}({\it{\Gamma}}). \end{align*} Proof. The idea of the proof is the same as in the proof of Lemma 2.11, using Lemma 3.9 instead of Lemma 2.10. □ 3.5 Proof of Theorem 3.1 First, we show boundedness and coercivity of $$a^s(\cdot,\cdot)$$ with respect to the norm $$| | |\cdot| | |$$. Then, the Lions–Stampacchia theorem (see, e.g., Glowinski et al., 1981; Rodrigues, 1987; Glowinski, 2008) proves unique solvability of (2.4), provided that $$F : U\to {\mathbb{R}}$$ is a linear functional, which follows by the boundedness of $${\it{\Theta}}:U\to{\mathbb{R}}$$. We start by showing the boundedness. Since $$b(\cdot,{\it{\Theta}}\cdot)$$ is symmetric and positive semidefinite, the Cauchy–Schwarz inequality proves \begin{align*}|b({\boldsymbol{u}},{\it{\Theta}}_\beta{\boldsymbol{v}})| \leq \beta b({\boldsymbol{u}},{\it{\Theta}}{\boldsymbol{u}})^{1/2} b({\boldsymbol{v}},{\it{\Theta}}{\boldsymbol{v}})^{1/2} = \beta \|{B{\boldsymbol{u}}}\|_{V'}\|{B{\boldsymbol{v}}}\|_{V'}\end{align*} for all $${\boldsymbol{u}},{\boldsymbol{v}}\in U$$. For the boundary terms, we consider \begin{align*}\varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq \varepsilon^{1/4}|\langle{\gamma_{\boldsymbol{n}}({\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| + \varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}|. \end{align*} The first term on the right-hand side is estimated with Lemma 3.10. Let $$\widetilde u \in H^1({\it{\Omega}})$$ be the quasi-harmonic extension of $$\gamma_0{\boldsymbol{u}}$$ and let $$\widetilde v \in H^1({\it{\Omega}})$$ be the quasi-harmonic extension of $$\gamma_0{\boldsymbol{v}}$$. Then, for the second term, we get with integration by parts (cf. Lemma 3.6), \begin{align*}\varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}} = \varepsilon^{-1/2}\Big( \varepsilon({\nabla \widetilde u} ,{\nabla\widetilde v}) + ({\widetilde u} ,{\widetilde v}) \Big) \leq \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} Together, this gives for the boundary term, \begin{align*}\varepsilon^{1/4} |\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{v}}}\rangle_{\it{\Gamma}}| \leq 3/\sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'}\varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})} + \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{v}}}\|_{H^{1/2}({\it{\Gamma}})}. \end{align*} The second boundary term $$\varepsilon^{1/4} \langle{\gamma_{\boldsymbol{n}}{\boldsymbol{v}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}$$ is treated identically. Altogether this proves the boundedness of $$a^s(\cdot,\cdot)$$. For the proof of coercivity we use Lemma 3.6, Lemma 3.10 and Young’s inequality to find that, for $$\delta>0$$, \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 &= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 = \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &= \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}(\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}-{\boldsymbol{u}})} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &\leq \|{B{\boldsymbol{u}}}\|_{V'}^2 + 3/\sqrt{2} \|{B{\boldsymbol{u}}}\|_{V'} \varepsilon^{-1/4}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})} + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}} \\ &\leq (1+\delta^{-1}\tfrac{9}4) \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}+\tfrac\delta2 \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} We choose $$\delta = \tfrac{9}8$$, which implies $$1+\delta^{-1}\tfrac{9}4 = 3$$. Then, subtracting the last term on the right-hand side, we obtain for $$\beta\ge 3$$, \begin{align*}\|{B{\boldsymbol{u}}}\|_{V'}^2 + \tfrac{7}{16} \varepsilon^{-1/2} \|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \leq 3 \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{1/4}\langle{\gamma_{\boldsymbol{n}}{\boldsymbol{u}}} ,{\gamma_0{\boldsymbol{u}}}\rangle_{\it{\Gamma}}\leq a^s({\boldsymbol{u}},{\boldsymbol{u}}) \quad\text{for all }{\boldsymbol{u}}\in U. \end{align*} Next we show (3.3). By Lemma 3.7 and the triangle inequality we obtain \begin{align*}\|{{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{{\boldsymbol{u}}-\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1}^2 + \|{\mathcal{E}\,\,\gamma_0{\boldsymbol{u}}}\|_{U,1}^2 \lesssim \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2. \end{align*} For the proof of the upper bound in (3.3) let $$\widetilde u\in H^1({\it{\Omega}})$$ be a function that attains the minimum in the definition of $$\|{\widehat u^b}\|_{1/2,{\mathcal{S}}}$$. Then, \begin{align*}\varepsilon^{-1/2} \|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 = \varepsilon^{-1/2}\big( \|{\widetilde u}\|_{}^2 + \varepsilon^{1/2}\|{\nabla \widetilde u}\|_{}^2 \big) \geq \varepsilon^{-1/2} \big( \|{\widetilde u}\|_{}^2 + \varepsilon\|{\nabla \widetilde u}\|_{}^2 \big). \end{align*} Since $$\widetilde u|_{\it{\Gamma}} = \widehat u^b|_{\it{\Gamma}} = \widehat u^a|_{\it{\Gamma}} = \gamma_0{\boldsymbol{u}}$$, the right-hand side is an upper bound of $$\varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2$$. Thus, together with Lemma 3.7, we get \begin{align*}| | |{{\boldsymbol{u}}}| | |^2 = \|{B{\boldsymbol{u}}}\|_{V'}^2 + \varepsilon^{-1/2}\|{\gamma_0{\boldsymbol{u}}}\|_{H^{1/2}({\it{\Gamma}})}^2 \lesssim \|{{\boldsymbol{u}}}\|_{U,2}^2 + \varepsilon^{-1/2}\|{\widehat u^b}\|_{1/2,{\mathcal{S}}}^2 \lesssim \|{{\boldsymbol{u}}}\|_{U,2}^2. \end{align*} The remainder of the proof follows the same arguments as in the proof of Theorem 2.1. 4 Examples In this section, we present various numerical examples in two dimensions ($$d=2$$). For the first example in Section 4.2 we take a manufactured solution $$u\in H^2({\it{\Omega}})$$. The standard finite element method with lowest-order discretization on quasi-uniform meshes converges at a rate $${\mathcal{O}}(h)$$, where $$h$$ denotes the diameter of elements in $${\mathcal{T}}$$. We observe the same optimal rate for the DPG methods with $$\star\in\{0,{\boldsymbol{n}},s\}$$, analysed in Section 2. In Section 4.3, we consider an L-shaped domain with unknown solution and an expected singularity at the reentrant corner. Indeed, we will see that a uniform method gives suboptimal convergence, whereas an adaptive method driven by the estimator $$\eta$$ from Section 2.4 recovers the optimal one. Finally, in Section 4.4, we use a family of manufactured solutions that exhibit typical boundary layers for the reaction-dominated diffusion problem. Our numerical results underline the robustness of the a posteriori error estimate, as stated by Theorem 3.4. 4.1 General setting As is usual for DPG methods we replace the infinite-dimensional test space $$V$$ used in the calculation of optimal test functions (1.5) by a finite-dimensional subspace $$V_h$$; that is, we replace the test function $${\it{\Theta}}_\beta{\boldsymbol{u}}_h$$ for $${\boldsymbol{u}}_h\in U_h$$ by $${\it{\Theta}}_{\beta,h}{\boldsymbol{u}}_h$$ defined through \begin{align*}\left( {{{\it{\Theta}}_{\beta,h}{\boldsymbol{u}}_h},{{\boldsymbol{v}}_h}} \right)_V = \beta b({\boldsymbol{u}}_h,{\boldsymbol{v}}_h) \quad\text{for all }{\boldsymbol{v}}_h\in V_h. \end{align*} Here we choose \begin{align*}V_h := \begin{cases}P^2({\mathcal{T}})\times[P^2({\mathcal{T}})]^2 & \text{for the methods from Section 2}, \\ P^2({\mathcal{T}})\times[P^2({\mathcal{T}})]^2 \times P^4({\mathcal{T}}) & \text{for the method from Section 3}. \end{cases}\end{align*} These choices are motivated by Gopalakrishnan & Qiu (2014). We refer the interested reader to this work for more details. The resulting DPG scheme is called the practical DPG method. For the scaling parameter of the test functions, we choose $$\beta=2$$ for the methods from Section 2 and $$\beta=3$$ for the method from Section 3. We use the standard basis for the lowest-order spaces $$U_h$$, i.e., the element characteristic functions for $$P^0({\mathcal{T}})$$, $$P^0({\mathcal{S}})$$, and nodal basis functions (hat functions) for $$S^1({\mathcal{S}})$$. These choices allow a simple implementation of the inequality constraints in the cones $$K_h^\star$$. We solve the discrete variational inequalities (2.6), (3.4) with a (primal–dual) active set algorithm (see Hoppe & Kornhuber, 1994; Kärkkäinen et al., 2003). More precisely, we implemented a modification of Kärkkäinen et al. (2003, Algorithm A1). (which deals with obstacle problems) for the present problem (here, we consider inequality constraints only for degrees of freedom that are associated with the boundary). For the problems where the solution is known in analytical form, we compute different error quantities depending on the underlying problem from Section 2 or 3. Section 2: Let $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)$$ denote the exact solution of (2.4) and let $${\boldsymbol{u}}_h=(u_h,{\boldsymbol\sigma}_h,\widehat u_h,\widehat\sigma_h)$$ be its approximation. We define \begin{alignat*}{2}\quad\qquad\text{err}(u) &:= {\|u-u_{h}\|}{}, &\quad \text{err}({\boldsymbol\sigma}) &:= \left\| {{{\boldsymbol\sigma}-{\boldsymbol\sigma}_h}} \right\|{}, \\ \quad\qquad\text{err}(\widehat u) &:= \left(\left\| {{u-\widetilde u_h}} \right\|{}^2 + \left\| {{\nabla(u-\widetilde u_h)}} \right\|{}^2\right)^{1/2}, &\quad \text{err}(\widehat\sigma) &:= \left(\left\| {{{\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h}} \right\|{}^2 + \left\| {{{{\rm div\,}}({\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h)}} \right\|{}^2\right)^{1/2}. \end{alignat*} Here, $$\widetilde u_h\in S^1({\mathcal{T}})$$ is the nodal interpolant of $$\widehat u_h$$ at the nodes of $${\mathcal{T}}$$. Similarly, $$\widetilde{\boldsymbol\sigma}_h$$ is the Raviart–Thomas interpolation of $$\widehat \sigma_h$$. Then, it follows by the definition of the trace norms, \begin{align*}\left\| {{{\boldsymbol{u}}-{\boldsymbol{u}}_h}} \right\|_{U} \leq \left(\text{err}(u)^2 + \text{err}({\boldsymbol\sigma})^2 + \text{err}(\widehat u)^2 + \text{err}(\widehat\sigma)^2\right)^{1/2}. \end{align*} Section 3: Let $${\boldsymbol{u}}=(u,{\boldsymbol\sigma},\rho,\widehat u^a,\widehat u^b,\widehat\sigma^a,\widehat\sigma^b)$$ denote the exact solution of (3.2) and let $${\boldsymbol{u}}_h=(u_h,{\boldsymbol\sigma}_h,\rho_h,\widehat u_h^a,\widehat u_h^b,\widehat\sigma_h^a,\widehat\sigma_h^b)$$ be its approximation. Define $$\text{err}(u)$$ and $$\text{err}({\boldsymbol\sigma})$$ as above and additionally \begin{align*}\text{err}(\widehat u^\star) &:= \left(\left\| {{u-\widetilde u_h^\star}} \right\|{}^2 + \varepsilon^{1/2}\left\| {{\nabla(u-\widetilde u_h^\star)}} \right\|{}^2\right)^{1/2}, \\ \text{err}(\widehat\sigma^\star) &:= \left(\left\| {{{\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h^\star}} \right\|{}^2 + \varepsilon\left\| {{{{\rm div\,}}({\boldsymbol\sigma}-\widetilde{\boldsymbol\sigma}_h^\star)}} \right\|{}^2\right)^{1/2}, \\ \text{err}(\rho) &:= \varepsilon^{1/2}\left\| {{\rho-\rho_h}} \right\|{}, \end{align*} for $$\star\in\{a,b\}$$, and $$\widetilde u_h^\star$$, $$\widetilde\sigma_h^\star$$ defined in the same way as above. Our total error estimator is \begin{align*}\text{err}({\boldsymbol{u}}) &:= \Big( \text{err}(u)^2 + \text{err}({\boldsymbol\sigma})^2 + \text{err}(\rho)^2 \\ &\quad +\varepsilon^{3/2} \text{err}(\widehat u^a)^2 + \varepsilon \text{err}(\widehat u^b)^2 + \varepsilon^{3/2}\text{err}(\widehat\sigma^a)^2+\varepsilon^{5/2}\text{err}(\widehat\sigma^b)^2 \Big)^{1/2}\end{align*} so that \begin{align*}\left\| {{{\boldsymbol{u}}-{\boldsymbol{u}}_h}} \right\|_{U,1} \leq \text{err}({\boldsymbol{u}}). \end{align*} For examples with singularities and/or boundary or interior layers, we use a standard adaptive algorithm that uses $$\eta(T)$$ and $$\eta(E)$$ to mark elements by the bulk criterion. For convenience, we define $$\eta({\mathcal{T}})$$ and $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ by \begin{align*}\eta({\mathcal{T}})^2 := \sum_{T\in{\mathcal{T}}} \eta(T)^2, \quad\eta({\mathcal{S}}_{\it{\Gamma}})^2 := \sum_{E\in{\mathcal{S}}_{\it{\Gamma}}} \eta(E)^2. \end{align*} 4.2 Piecewise smooth solution (Section 2) We consider the domain $${\it{\Omega}}:=(-1,1)\times (0,1)$$ and the manufactured solution \begin{align*}u(x,y) := \begin{cases}-16x^2(1-x)y(1-y), & x\geq 0, \\ 2(x+1)^3 - 3(x+1)^2 +1, & x<0. \end{cases}\end{align*} This solution satisfies $$u(x,y) = 0$$ on the part of $${\it{\Gamma}}=\partial{\it{\Omega}}$$ where $$x\geq 0$$ and $$\partial_{{\boldsymbol{n}}_{\it{\Omega}}}u = 0$$ on the part of $${\it{\Gamma}}$$ where $$x<0$$. We calculate $$f:=-{\it{\Delta}} u + u$$ and note that $$u\in H^1({\it{\Delta}},{\it{\Omega}})$$. Also note that $$u(x,y)$$ is smooth in both regions $$x>0$$ and $$x<0$$. Moreover, observe that $$u$$ satisfies the Signorini problem (1.1) with $$c=1$$. Our initial mesh consists of $$8$$ congruent triangles. We solve (2.6) for $$\star \in \{0,{\boldsymbol{n}},s\}$$ and plot the errors $$\text{err}(u)$$, $$\text{err}({\boldsymbol\sigma})$$, $$\text{err}(\widehat u)$$ and $$\text{err}(\widehat\sigma)$$ for a sequence of uniformly refined triangulations. Moreover, in the case $$\star=s$$, we compare these error quantities with the reliable error estimator $$\eta$$. The results are given in Fig. 1 for $$\star=0$$, $$\star={\boldsymbol{n}}$$ and Fig. 2 for $$\star=s$$. We observe optimal convergence rates $${\mathcal{O}}(h^\alpha) = {\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ with $$\alpha=1$$ for the error quantities. This rate is visualized by a triangle. In Fig. 2, we see that also the estimator $$\eta({\mathcal{T}})$$ converges with this rate whereas $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ has a higher convergence rate of approximately $$\alpha = 2.8$$. Fig. 1. View largeDownload slide Error quantities for the example from Section 4.2 with $$\star=0$$ (left) and $$\star={\boldsymbol{n}}$$ (right). Fig. 1. View largeDownload slide Error quantities for the example from Section 4.2 with $$\star=0$$ (left) and $$\star={\boldsymbol{n}}$$ (right). Fig. 2. View largeDownload slide Error quantities and estimators for the example from Section 4.2 with $$\star=s$$. Fig. 2. View largeDownload slide Error quantities and estimators for the example from Section 4.2 with $$\star=s$$. 4.3 Unknown solution (Section 2) Let $${\it{\Omega}} = (-1,1)^2\setminus [-1,0]^2$$ with initial triangulation visualized in Fig. 3. We define Fig. 3. View largeDownload slide L-shaped domain with initial triangulation of $$12$$ elements. Fig. 3. View largeDownload slide L-shaped domain with initial triangulation of $$12$$ elements. \begin{align*}f(x,y) := \begin{cases}-1, & |(x,y)|\leq 0.8, \\ \tfrac12 & \text{otherwise}. \end{cases}\end{align*} For this right-hand side, the solution is not known to us in analytical form. Therefore, we compute only the error estimators. The results are plotted in Fig. 4. We observe that uniform refinement leads to a reduced order of convergence $${\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ of approximately $$\alpha=0.7$$, whereas adaptive refinement regains the optimal order $$\alpha=1$$. This is a strong indicator that the unknown solution has a singularity at the reentrant corner, which is what one expects. Figure 5 visualizes meshes at different steps of the adaptive loop and supports this observation. Fig. 4. View largeDownload slide Estimators for the example from Section 4.3 for uniform and adaptive refinement. Fig. 4. View largeDownload slide Estimators for the example from Section 4.3 for uniform and adaptive refinement. Fig. 5. View largeDownload slide Meshes at different steps in the adaptive algorithm for the problem from Section 4.3. Fig. 5. View largeDownload slide Meshes at different steps in the adaptive algorithm for the problem from Section 4.3. 4.4 Piecewise smooth solution with boundary layer (Section 3) Let $${\it{\Omega}}:=(-1,1)\times (0,1)$$ with manufactured solution \begin{align*}u(x,y) := \begin{cases}-x^2\left(e^{-2(1-x)/\sqrt\varepsilon}\right) \left( e^{-y/\sqrt\varepsilon} + e^{-(1-y)/\sqrt\varepsilon} - e^{-1/\sqrt\varepsilon} - 1\right), &x\geq 0, \\ 2(x+1)^3 - 3(x+1)^2 +1, & x<0. \end{cases}\end{align*} We choose $$f=-\varepsilon{\it{\Delta}} u+u$$. Observe that $$u$$ satisfies (1.1) with $$c=\varepsilon$$. In particular, $$u$$ has a layer of order $$\sqrt{\varepsilon}$$ at the boundary for $$x\geq 0$$. We solve the variational inequality (3.4) on a sequence of adaptively refined triangulations for $$\varepsilon\in\{10^{-2},10^{-4},10^{-6},10^{-8}\}$$. We start with a coarse initial triangulation that consists of only $$\#{\mathcal{T}}_0 = 8$$ congruent triangles. In Fig. 6, we compare the estimator $$\eta$$ with the total error $$\text{err}({\boldsymbol{u}})$$. As in the works Heuer & Karkulik (2017) and Führer & Heuer (2016), we observe that, after boundary layers have been resolved, our method leads to an optimal convergence rate $${\mathcal{O}}( (\#{\mathcal{T}})^{-\alpha/2})$$ ($$\alpha=1$$). In the pre-asymptotic range, we see that there is a small gap between the curves representing $$\eta({\mathcal{T}})$$ and $$\text{err}({\boldsymbol{u}})$$. This is due to the fact that the optimal test functions cannot be approximated accurately enough in the test space $$V_h$$. Similar observations have been made in Heuer & Karkulik (2017) and Führer & Heuer (2016). After layers are resolved we see that the curves are close together and uniformly in $$\varepsilon$$ (for the selected values). This confirms the robustness of the a posteriori estimate by Theorem 3.4. We also see that the boundary estimator $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ is small in comparison with $$\eta({\mathcal{T}})$$, and has a higher convergence rate. Fig. 6. View largeDownload slide Comparison of total error $$\text{err}({\boldsymbol{u}})$$ and estimators $$\eta({\mathcal{T}})$$, $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ for the problem from Section 4.4. Fig. 6. View largeDownload slide Comparison of total error $$\text{err}({\boldsymbol{u}})$$ and estimators $$\eta({\mathcal{T}})$$, $$\eta({\mathcal{S}}_{\it{\Gamma}})$$ for the problem from Section 4.4. Our primary interest was to construct a robust method in the sense that the error in the balanced norm of the field variables $$u,{\boldsymbol\sigma},\rho$$ is controlled uniformly in $$\varepsilon$$. The error estimator $$\eta$$ does precisely this, as stated by Theorem 3.4 and seen in Fig. 6. To further underline this statement, Fig. 7 shows the ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$. We observe that, again after boundary layers have been resolved, this ratio is between $$0.5$$ and $$0.55$$ uniformly with respect to $$\varepsilon$$. In fact, this number is close to $$1/\sqrt{\beta} = 1/\sqrt{3}\sim 0.5774$$. Note that, by the product structure of $\( \left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'}^2 = \sum_{T\in{\mathcal{T}}} \left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'(T)}^2,\)$ so that $$\eta({\mathcal{T}})=\sqrt{\beta}\left\| {{B{\boldsymbol{u}}_h-L}} \right\|_{V'}$$; cf. (2.7). Our numerical results therefore indicate that the slight overestimation of the error by $$\eta$$ is due to the choice of $$\beta$$, and is (asymptotically) uniform in $$\varepsilon$$. Fig. 7. View largeDownload slide Ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$ for the problem from Section 4.4. Fig. 7. View largeDownload slide Ratio $$(\text{err}(u)^2+\text{err}({\boldsymbol\sigma})^2+\text{err}(\rho)^2)^{1/2}/\eta({\mathcal{T}})$$ for the problem from Section 4.4. Funding CONICYT through FONDECYT projects (1150056 and 3150012). References Attia ,F. S. , Cai ,Z. & Starke ,G. ( 2009 ) First-order system least squares for the Signorini contact problem in linear elasticity. SIAM J. Numer. Anal. , 47 , 3027 – 3043 . Google Scholar Crossref Search ADS Brezzi ,F. , Hager ,W. W. & Raviart ,P.-A. ( 1977 ) Error estimates for the finite element solution of variational inequalities . Numer. Math. , 28 , 431 – 443 . Google Scholar Crossref Search ADS Brezzi ,F. , Hager ,W. W. & Raviart ,P.-A. ( 1978 ) Error estimates for the finite element solution of variational inequalities. II. Mixed methods . Numer. 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Journal

IMA Journal of Numerical AnalysisOxford University Press

Published: Oct 16, 2018

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