A time-dependent wave-thermoelastic solid interaction

A time-dependent wave-thermoelastic solid interaction Abstract This paper presents a combined field and boundary integral equation method for solving the time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid–structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analysed with Lubich’s approach for time-dependent boundary integral equations. Existence and uniqueness results are established in terms of time-domain data with the aid of Laplace domain techniques. Galerkin semidiscretization approximations are derived and error estimates are obtained. A full discretization based on the convolution quadrature method is also outlined. Some numerical experiments in two dimensions are also included in order to demonstrate the accuracy and efficiency of the procedure. 1. Introduction The mathematical study of the thermodynamic response of a linearly elastic solid to mechanical strain dates back at least to Duhamel’s (1837) pioneering work on thermoelastic materials where he proposed the constitutive relation linking the temperature variations and elastic strains with the thermoelastic stress now known as the Duhamel–Neumann law (Carlson, 1972; Maugin, 2014). Kupradze’s (1979) encyclopedic works can be considered the standard reference for a modern mathematical treatment of the purely thermoelastic problem. The dynamic problem is dealt with in more recent works like Ortner & Wagner (1992) and Wagner (1994) where the matrix of fundamental solutions for the dynamic equations is revisited, while Jakubowska (1982, 1984) provide generalized Kirchhoff-type formulas for thermoelastic solids. In the case of the scattering of thermoelastic waves, major theoretical contributions have been made by Çakoni & Dassios (1998). The unique solvability of a boundary integral formulation is established in the study by Çakoni (2000) and the interaction of elastic and thermoelastic waves is explored for homogeneous materials in the study by Dassios & Kostopoulos (1994). The study of time-harmonic interaction between a scalar field and a thermoelastic solid has been the subject of works like Lopat’ev (1979) where the interface is taken to be a plane, or Lin & Raptis (1983) and Jentsch & Natroshvili (1997) where time-harmonic scattering by bounded obstacles is considered. In this paper we present a combined field and boundary integral method for a time-dependent fluid–thermoelastic solid interaction problem. The approach here is a generalization of the method employed by the authors for treating time-dependent fluid–structure interaction problems in Hsiao et al. (2016) and Hsiao et al. (2013). The present communication is an improvement over those previous efforts in the sense that it considers a more general constitutive law that accounts for the coupling between elastic and thermal effects. To our knowledge, other than the preliminary results in Sánchez-Vizuet (2016), no attempt has been made to investigate with rigorous justifications the time-dependent acoustic scattering by a thermoelastic obstacle. The setting is introduced in Section 2, along with the physical assumptions and the constitutive relation under consideration leading to the time-domain system of governing equations. The problem is then recast in Section 3, where the Laplace domain system is transformed into an equivalent integro-differential nonlocal problem that will be formulated variationally for discretization later on. The question of existence and uniqueness of the solutions to the nonlocal problem is dealt with in Section 4. The error analysis of the proposed discretization is addressed in Section 5, where semidiscrete error estimates for spatial discretization are provided. The final Section 6 discusses the computational considerations related to the numerical solution of the discrete problem. A full discretization using convolution quadrature (CQ) in time is outlined and the coupling of boundary and finite elements for the spatial discretization is discussed. Convergence experiments in two dimensions are performed for test problems in both frequency and time domains as a demonstration of the applicability of the formulation, which remains valid also in three dimensions. For time discretization, both second-order backward differentiation formula (BDF2) and trapezoidal rule CQ are used, providing evidence that the approximation is stable and of second order globally. One simple illustrative experiment in the time domain using the proposed formulation is included to demonstrate the applicability of the procedure. In closing, we remark that for a homogeneous thermoelastic solid medium, a pure boundary integral equation formulation may be adapted as in the fluid–structure interaction problem (Hsiao et al., 2016). We will pursue these investigations in a separate communication. 2. Formulation of the problem Consider a thermoelastic solid with constant density $$\rho _{\varSigma }$$ in an undeformed reference configuration and at thermal equilibrium at temperature $$\varTheta _{0}$$. Under the action of external forces the body will be subject to internal stresses that will induce local variations of temperature. Reciprocally, if a heat source induces a change in temperature, the body will react by dilating or contracting and this will create internal stresses and deformations. We will denote by U the elastic deformation with respect to the reference configuration and by $$\varTheta $$ the variation of temperature with respect to the equilibrium temperature. In the classical linear theory (Kupradze, 1979; Landau et al., 1986), the coupling between the mechanical strain and the thermal gradient is modeled by the Duhamel–Neumann law. This constitutive relation defines the thermoelastic stress$$\boldsymbol \sigma (\mathbf U,\varTheta )$$ and the thermoelastic heat flux$$\mathbf F(\mathbf U,\varTheta )$$ (also known as free energy) as functions of the elastic displacement and the variation in temperature through the equations  \begin{align*} \boldsymbol\sigma :=\,& \mathbf C\varepsilon(\mathbf U) - \zeta\varTheta\mathbf I\,,\\ \mathbf F :=\,& -\eta\,\frac{\partial \mathbf U}{\partial t} + \kappa \nabla\varTheta. \end{align*} In the previous expressions,   $$ \varepsilon (\mathbf{U}) := \tfrac{1}{2}\left( \nabla\mathbf{U} + \left(\nabla\mathbf{U}\right)^{\textrm T}\right) $$is the elastic strain tensor, I is the 3 × 3 identity matrix, $$\kappa $$ is the thermal diffusivity coefficient, which from physical principles (Hahn & Ozisik, 2012) is required to be positive, $$\zeta $$ is the product of the volumetric thermal expansion coefficient and the bulk modulus of the material and $$\eta $$ is given by the relation   $$ \eta = \varTheta_{0}\zeta/c_{\textrm{vol}}. $$Here the volumetric heat capacity $$c_{\textrm{vol}}$$ is the ratio between the thermal diffusivity and the thermal conductivity, and it can also be expressed as the product of the mass density and the specific heat capacity. For the case of a homogeneous isotropic material that we are considering, the elastic stiffness tensorC is given by   $$ \mathbf C_{ijkl}:= \lambda\delta_{ij}\delta_{kl} +\mu\left(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}\right), $$where the constants $$\lambda $$ and $$\mu $$ are Lamé’s second parameter and the shear modulus, respectively, and $$\delta _{ij}$$ is Kronecker’s delta. We are concerned with a time-dependent direct scattering problem in fluid–thermoelastic solid interaction, which can be simply described as follows: an acoustic wave propagates in a fluid domain of infinite extent in which a bounded thermoelastic body is immersed. Throughout the paper, we let $$\varOmega _{-}$$ be the bounded domain in $$\mathbb{R}^{3}$$ occupied by the thermoelastic body with a Lipschitz boundary $$\varGamma $$ and we let $$\varOmega _{+} := \mathbb{R}^{3} \setminus \overline{\varOmega }_{-}$$ be its exterior, occupied by a compressible fluid. The problem is then to determine the scattered velocity potential V in the fluid domain, the deformation of the solid U and the variation of the temperature $$\varTheta $$ in the obstacle. It is assumed that $$|\varTheta /\varTheta _{0}|\ll1$$. The governing equations of the displacement field U and temperature field $$\varTheta $$ are the thermo-elastodynamic equations:  \begin{align} \rho_{\varSigma} \frac{\partial^{2}\mathbf{U}}{\partial t^{2}} - \varDelta^{*} \mathbf{U} + \zeta\, \nabla\varTheta = \mathbf{0} \quad \textrm{in}\; \varOmega_{-} \times (0,T), \end{align} (2.1)  \begin{align} \frac{1}{\kappa}\frac{\partial \varTheta}{\partial t} - \varDelta \varTheta + \eta\; \frac{\partial}{\partial t}(\nabla\cdot\mathbf{U}) = 0 \quad \textrm{in}\; \varOmega_{-} \times (0,T), \end{align} (2.2)where T is a given positive final time, and as usual the symbol $$\varDelta ^{\ast }$$ is the Lamé operator defined by   $$ \varDelta^{\ast} \mathbf{U} := \mu \varDelta \mathbf{U} + (\lambda + \mu) \nabla(\nabla\cdot \mathbf U). $$We remark that if the thermal effect is neglected ($$\zeta =0$$), Duhamel–Neumann’s law reduces to the usual expression for Hooke’s law of the classical theory for an arbitrary isotropic medium (see, e.g., Carlson, 1972 and Kupradze, 1979). In the thermoelastic medium, the given physical constants $$ \rho _{\varSigma }, \lambda , \mu , \zeta , \eta , \kappa $$, are assumed to satisfy the inequalities   $$ \rho_{\varSigma}> 0, \;\;\mu > 0, \;\;3 \lambda + 2 \mu > 0, \;\;\frac{\zeta}{\eta} > 0, \;\;\kappa > 0. $$In the fluid domain $$\varOmega _{+}$$, we consider a barotropic and irrotational flow of an inviscid and compressible fluid with density $$\rho _{f}$$ as in the study by Hsiao et al. (2013). The formulation can be presented in terms of a scalar potential V = V (x, t) such that the scattered velocity field V and the pressure P are given by   $$ \mathbf{V} = - \nabla\; V \quad\textrm{and}\quad P={\rho_{f}} \frac{\partial V}{\partial t}. $$We thus arrive at the wave equation   \begin{align} \frac{1}{c^{2}} \frac{\partial^{2} V}{\partial t^{2}} - \varDelta V = 0 \quad\textrm{in}\quad \varOmega_{+} \times (0, T), \end{align} (2.3)where c is the sound speed. On the interface $$\varGamma $$ between the solid and the fluid we have the transmission conditions   \begin{align} \boldsymbol\sigma(\mathbf U, \varTheta)^{-}\mathbf{n} =\, -{\rho_{f}}\;\left( \frac{\partial V}{\partial t} + \frac{\partial V^{\textrm{inc}}}{\partial t}\right)^{+}\mathbf{n} \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4a)  \begin{align} \frac{ \partial\mathbf{U}^{-}}{\partial t} \cdot\mathbf{n} = \, - \left(\frac{\partial V}{\partial n} + \frac{\partial V^{\textrm{inc}}}{\partial n}\right)^{+} \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4b)  \begin{align} \frac{\partial \varTheta}{\partial n}^{-} = \, 0 \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4c)where n is the exterior unit normal to $$\varOmega _{-}$$, and $$V^{\textrm{inc}} $$ denotes the given incident field, which is assumed to be supported away from $$\varGamma $$ at t = 0. Here and in the sequel, we adopt the notation that $$q^{\mp }$$ denotes the limit of the function q on $$\varGamma $$ from $$\varOmega _{\mp }$$, respectively. Regarding the transmission conditions we remark that from the physical point of view, equation (2.4a) enforces the equilibrium of pressure at the solid–fluid interface, condition (2.4b) expresses the continuity of the normal component of the velocity field, and (2.4c) refers to a thermally insulated body. We assume the causal initial conditions   \begin{align} \mathbf{U}(x,t) = \frac{\partial\mathbf{U}(x,t)}{\partial t} =\mathbf{0}, \quad \varTheta(x, t) = 0 \quad \textrm{for}\quad x \in \varOmega_{-},\;\; t \leq 0, \end{align} (2.5a)  \begin{align} V(x,t) = \frac{\partial V}{\partial t} (x,t) = 0\;\; \textrm{for} \quad x\in \varOmega_{+},\;\; t \leq 0. \end{align} (2.5b)We will study the time-dependent scattering problem consisting of the partial differential equations (2.1)--(2.3) together with the transmission conditions (2.4a)–(2.4c) and the homogeneous initial conditions (2.5a)–(2.5b). 3. Reduction to a nonlocal problem on a bounded domain In order to apply Lubich’s approach as in the case of fluid–structure interaction (Hsiao et al., 2013; Hsiao et al., 2016), we first need to transform the initial–boundary transmission problem (2.1)–(2.4c) in the Laplace domain. To do that we first reduce the corresponding problem to a nonlocal boundary value problem. We begin with the Laplace transform for a restricted class of distributions. Let X be a Banach space and $$\mathcal S(\mathbb R)$$ denote the Schwartz class of functions. We say that $$F: \mathcal S(\mathbb R)\rightarrow X$$ is a causal tempered distribution with values in X if it is a continuous linear map such that   $$ F(\varphi) = 0 \quad \forall\, \varphi\in\mathcal S (\mathbb R) \textrm{ such that } \textrm{supp}\;\varphi \subset (-\infty,0). $$For such a distribution and for   $$ s\in\mathbb C_{+} :=\big\{ s \in \mathbb C : \textrm{Re}\;s> 0\big\}, $$the Laplace transform of F can be defined in a natural way by   $$ f(s)= \mathcal{L}\big\{F\big\}(s) := \int_{0}^{\infty} e^{-st} F(t)\, \textrm{d}t, $$where the integral must be understood in the sense of Bochner (Sayas, 2016) when F is integrable or as a duality product in the general case. We remark that the Laplace transform can be defined for a much broader class of distributions (Beltrami & Wohlers, 1966; Dautray & Lions, 1992), but this restricted class suffices for the current application. Then let   $$ \mathbf{u}:= \mathbf{u}(x,s)= \mathcal{L}\big\{\mathbf{U}(x,t)\big\}, \;\; \theta:=\theta(x,s) = \mathcal{L}\big\{{\varTheta(x,t)} \big\},\;\; v:=v(x,s) = \mathcal{L}\big\{V(x,t)\big\}. $$The initial–boundary transmission problem consisting of (2.1)–(2.4c) in the Laplace transformed domain becomes the following transmission boundary value problem:   \begin{align} {\hskip-76pt} - \varDelta^{*} \mathbf{u} + \rho_{\varSigma} s^{2} \mathbf{u} + \zeta \nabla\theta = \, \mathbf{0} \quad \textrm{in}\quad \varOmega_{-}, \end{align} (3.1a)  \begin{align} {\hskip-77.5pt} - \varDelta \theta + \frac{s}{\kappa} ~\theta + s~ \eta ~\nabla\cdot \mathbf{u} =\, 0 \quad \textrm{in} \quad \varOmega_{-}, \end{align} (3.1b)  \begin{align}{\hskip-31.5pt} -\varDelta v + \frac{s^{2}}{c^{2}} \;v = \, 0 \quad \textrm{in}\quad \varOmega_{+}, \end{align} (3.1c)  \begin{align}{\hskip-31pt} \boldsymbol{\sigma} (\mathbf{u}, \theta )^{-} \mathbf{n} + \rho_{f}\; s v^{+} \mathbf{n} =\, - \rho_{f}s v^{\textrm{inc}}\mathbf{n} \quad \textrm{on}\quad \varGamma, \end{align} (3.1d)  \begin{align} {\hskip-19pt} s \; \mathbf{u}^{-} \cdot \mathbf{n} + \frac{ \partial v}{\partial n}^{+} =\, -\frac{\partial v^{\textrm{inc}}}{\partial n} \quad \textrm{on}\quad \varGamma, \end{align} (3.1e)  \begin{align} \frac{\partial \theta}{\partial n}^{-} = \, 0 \quad \textrm{on} \quad \varGamma. \end{align} (3.1f)We remark that (3.1) is an exterior scattering problem for which normally a radiation condition is needed in order to guarantee the uniqueness of the solution. However, in the present case no additional radiation condition is required and global $$H^{1}$$ behavior at infinity suffices; uniqueness then follows from a standard energy argument and the fact that Re s > 0. To derive a proper nonlocal boundary problem, as usual, we begin via Green’s third identity with the representation of the solutions of (3.1c) in the form:   \begin{align} v = \mathcal{D}(s){\phi} - \mathcal{S}(s){\lambda} \quad \textrm{in} \quad \varOmega_{+}, \end{align} (3.2)where $${\phi }:= v^{+}$$ and $${\lambda }:= \partial v^{+} /\partial n$$ are the Cauchy data for v in (3.1c) and $$\mathcal{S}(s)$$ and $$\mathcal{D}(s)$$ are the simple-layer and double-layer potentials, respectively, defined by   \begin{align} \mathcal{S}(s){\lambda} (x) :=\, \int_{\varGamma} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y},\qquad x \in \mathbb R^{3}\setminus\varGamma, \end{align} (3.3)  \begin{align} \mathcal{D}(s){\phi} (x) :=\, \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y},\quad x \in \mathbb R^{3}\setminus\varGamma, \end{align} (3.4)with respect to the complex frequency $$s\in \mathbb C_{x+}$$, where   $$ E_{s/c}(x,y) = \frac{e^{- s/c\; |x-y|}}{4 \pi |x-y|} $$is the fundamental solution of the operator in (3.1c). By standard arguments in potential theory, we have the relations for the Cauchy data $${\lambda }$$ and $${\phi } $$,   \begin{align} \left(\begin{array}{cccccc} {\phi} \\[3mm] {\lambda}\\ \end{array}\right) = \left ( \begin{matrix} \frac{1}{2}I + K(s) & -V(s) \\[3mm] -W(s) & \frac{1}{2}I - K^{\prime}(s) \\ \end{matrix} \right )\left(\begin{array}{cccccc} {\phi} \\[3mm] {\lambda}\\ \end{array}\right) \quad \textrm{on} \quad \varGamma. \end{align} (3.5)Here $$V, K, K^{\prime } $$ and W are the four basic boundary integral operators familiar from potential theory (Hsiao & Wendland, 2008) such that   \begin{align*} V(s) \lambda (x) :=\,& \int_{\varGamma} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\quad\ \ x \in \varGamma,\\ K(s){\phi} (x) :=\,& \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\! x \in \varGamma,\\ K^{\prime}(s)\lambda (x) :=\,& \int_{\varGamma} \frac{\partial}{\partial n_{x}} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\! x \in \varGamma,\\ W(s){\phi} (x) :=\,&- \frac{\partial}{\partial n_{x}} \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y}, \qquad x \in \varGamma. \end{align*}The operator $$K^{\prime }(s)$$ is the real transpose of K(s), i.e., it is the conjugate of the Hilbert space adjoint. Note that all our Hilbert spaces are complexified versions of real Hilbert spaces and real adjoints/transposes can be defined. By using the transmission condition (3.1e), we obtain from the second boundary integral equation in (3.5),   \begin{align} -s \mathbf{u}^{-} \cdot \mathbf{n} + W(s){\phi} - \left( \tfrac{1}{2} I - K^{\prime}(s)\right) {\lambda} = \frac{\partial v^{\textrm{inc} }}{\partial n} \;\quad \textrm{on}\quad \varGamma, \end{align} (3.6)while the first boundary integral equation in (3.5) is simply  \begin{align} \left( \tfrac{1}{2}I - K(s)\right){\phi} + V(s){\lambda} = 0 \quad \textrm{on} \quad \varGamma. \end{align} (3.7)With the Cauchy data $$\phi $$ and $$\lambda $$ as new unknowns, the partial differential equation (3.1c) in $$\varOmega _{+}$$ may be eliminated. This leads to a nonlocal boundary value problem in $$\varOmega _{-}$$ for the unknowns $$(\mathbf u, \theta , \phi , \lambda )$$ satisfying the partial differential equations (3.1a), (3.1b), and the boundary integral equations (3.6), (3.7) together with the conditions (3.1d) and (3.1f) on $$\varGamma $$. Here and in the sequel let $$\gamma ^{\mp } $$ and $$\partial _{n}^{\mp }$$ denote trace operators of the functions and their normal derivatives from inside and outside $$\varGamma $$, respectively. We will use the symbol $$(\cdot ,\cdot )_{\mathcal{O}}$$ interchangeably to denote the scalar, vector or matrix $$L^{2}$$ inner products of functions defined on the open set $$\mathcal O$$, while the angled brackets $$\langle \cdot ,\cdot \rangle _{\varGamma }$$ will be reserved for pairings between elements of the trace space and its dual. All the forms will be kept bilinear and conjugation will be done explicitly when needed. Finally, the space $$\mathbf H^{1}(\mathcal O)$$ should be understood as the Cartesian product of copies of the standard scalar Sobolev space $$H^{1}(\mathcal O)$$ endowed with the natural product norm. Let us first consider the unknowns $$(\mathbf u,\theta ) \in{\mathbf H}^{1}(\varOmega _{-})\times H^{1}(\varOmega _{-})$$. Multiplying (3.1a) by the testing function v and integrating by parts, we obtain the weak formulation of (3.1a):   \begin{align} a({\mathbf u},{\mathbf v}; s) - \langle{\boldsymbol \sigma}(\mathbf u,\theta)\mathbf n, \gamma^{-}{\mathbf v}\rangle_{\varGamma} - \zeta(\theta, \; \nabla\cdot\mathbf v)_{\varOmega_{-}} = 0, \end{align} (3.8)where   \begin{align} a(\mathbf u, \mathbf v; s): = \left( \mathbf C\boldsymbol \varepsilon (\mathbf u), \boldsymbol \varepsilon ( \mathbf v)\right)_{\varOmega_{-}} + s^{2}\rho_{\varSigma}( \mathbf u, \mathbf v)_{\varOmega_{-}}. \end{align} (3.9)In terms of the transmission condition (3.1d), we obtain from (3.8),   \begin{align} a({\mathbf u},{\mathbf v}; s) - \zeta (\theta, \; \nabla\cdot\mathbf v)_{\varOmega_{-}} + \rho_{f} s \langle \phi~{\mathbf n}, \gamma^{-}{\mathbf v}\rangle_{\varGamma} = - s\rho_{f} \langle v^{\textrm{inc}} \mathbf n, \gamma^{-} \mathbf v \rangle_{\varGamma}. \end{align} (3.10)Similarly, multiplying (3.1b) by the test function $$\vartheta $$, integrating by parts and making use of the condition (3.1f), we have   \begin{align} b( \theta, \vartheta; s) + s\; \eta(\nabla\cdot\mathbf u, \vartheta)_{\varOmega_{-}}= 0 \end{align} (3.11)with   \begin{align} b(\theta, \vartheta;s) := (\nabla \theta, \nabla \vartheta)_{\varOmega_{-}} + \frac{s}{\kappa} ( \theta, \vartheta)_{\varOmega_{-}}. \end{align} (3.12) Now let   $$\begin{aligned} \mathbf A_{s}:\;& \mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (\mathbf H^{1}(\varOmega_{-}))^{\prime}\!, \\ &\;\mathbf u \;&\; \longmapsto \;&\; a(\mathbf u, \,\cdot\,;s), \\ B_{s} :\;& H^{1}(\varOmega_{-})\;& \;\longrightarrow\;&\; (H^{1}(\varOmega_{-}))^{\prime}\!, \\ & \; \theta \;&\; \longmapsto \,& \; b(\theta,\,\cdot\,;s) \end{aligned}$$ be the operators associated to the bilinear forms (3.9) and (3.12), respectively, and consider similarly the operators   $$\begin{aligned} \textrm{div}:\; &\mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (\mathbf H^{1}(\varOmega_{-}))^{\prime}\!, \\ &\;\mathbf u \;&\; \longmapsto \;&\; (\nabla\cdot\mathbf u,\,\cdot\,)_{\varOmega_{-}}, \\ \gamma_{n}:\; & \mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (H^{1/2}(\varGamma))^{\prime}=H^{-1/2}(\varGamma), \\ &\;\mathbf u \;&\; \longmapsto \;&\; \langle\gamma^{-}\mathbf u\cdot\mathbf n,\,\cdot\,\rangle_{\varGamma}. \end{aligned}$$ The operator div is the distributional divergence operator followed by the canonical injection of $$L^{2}(\varOmega _{-})\equiv L^{2}(\varOmega _{-})^{\prime }$$ into the dual space of $$H^{1}(\varOmega _{-})$$. From (3.10), (3.12), (3.6) and (3.7), the nonlocal problem may be formulated as a system of operator equations: given data $$(d_{1}, d_{2}, d_{3}, d_{4}) \in X^{\prime },$$ find $$(\mathbf u, \theta , \phi , \lambda ) \in X $$ such that   \begin{align} \pmb{\mathcal{A}} \left(\begin{array}{c} \mathbf{u} \\ \theta \\ \phi\\ \lambda \\ \end{array}\right):= \left ( \begin{array}{cccc} \mathbf A_{s}& -\zeta~\textrm{div}^{\prime} & s ~\rho_{f}\; \gamma_{n}^{\prime} & 0 \\ s ~\eta ~\textrm{div} & B_{s} &0&0 \\ - s~ \gamma_{n} & 0 & W(s) & \!\!\! - \frac{1}{2} I + K^{\prime}(s) \\ 0 & 0 & \!\!\! \frac{1}{2}I - K(s) & V(s)\\ \end{array} \right ) \left(\begin{array}{c} \mathbf{u} \\ \theta\\ \phi\\ \lambda \\ \end{array}\right) = \left(\begin{array}{c} d_{1}\\ d_{2}\\ d_{3}\\ d_{4} \end{array}\right). \end{align} (3.13)In the above expression data is given by   \begin{align} d_{1} = -s~ \rho_{f}\; \gamma_{n}^{\prime}\gamma^{+}v^{\textrm{inc}}, \quad d_{2} = 0, \quad d_{3} = {\partial_{n}^{+} v^{\textrm{inc}}}, \quad d_{4} = 0. \end{align} (3.14)We have made use of the product spaces   \begin{align*} X := \,&{\mathbf H}^{1}( \varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^{1/2}(\varGamma) \times H^{-1/2}(\varGamma), \\ X^{\prime} :=\,& ({\mathbf H}^{1}(\varOmega_{-}))^{\prime} \times (H^{1}(\varOmega_{-}) )^{\prime} \times H^{-1/2}(\varGamma) \times H^{1/2}(\varGamma). \end{align*}Our aim is to show that equation (3.13) has a unique solution in X. We will do this in the next section. 4. Existence and uniqueness results Before considering the existence and uniqueness results, we first discuss the invertibility of the operator $$\pmb{\mathcal{A} }$$ in (3.13). We begin with the definitions of the following energy norms:   \begin{align} |\!|\!|\mathbf u|\!|\!|_{|s|, \varOmega_{-}}^{2} :=\, ( \mathbf C\boldsymbol\varepsilon({\mathbf u}), \boldsymbol{\varepsilon} (\bar{\mathbf u} ) )_{\varOmega_{-}} + \rho_{\varSigma} \| s \; \mathbf u \|^{2}_{\varOmega_{-}},\quad \mathbf u \in{\mathbf H}^{1}(\varOmega_{-}), \end{align} (4.1)  \begin{align} |\!|\!|\theta|\!|\!|^{2}_{|s|, \varOmega_{-}} :=\, \| \nabla \theta\|^{2}_{\varOmega_{-}} + \kappa^{-1} \| s^{1/2} \; \theta \|_{\varOmega_{-}}^{2},\quad \theta \in H^{1}(\varOmega_{-}), \end{align} (4.2)  \begin{align} |\!|\!|v|\!|\!|^{2}_{|s|, \varOmega_{+}} :=\, \| \nabla v\|^{2}_{\varOmega_{+}} + c^{-2} \| s \;v \|^{2}_{\varOmega_{+}},\quad v \in H^{1}(\varOmega_{+}). \end{align} (4.3)For the complex Laplace parameter s we will denote   $$ \sigma:= \textrm{Re}\;s, \quad \underline{\sigma}:= \min\{1,\sigma\}, $$and will make use of the following equivalence relations for the norms:   \begin{align} \underline{\sigma }|\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega_{-}} \leq |\!|\!|\mathbf{u}|\!|\!|_{|s|, \varOmega_{-}}\leq \frac{|s|}{\underline{\sigma}} |\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega_{-}}, \end{align} (4.4)  \begin{align} \sqrt{\underline{\sigma}}|\!|\!|\theta|\!|\!|_{1, \varOmega_{-}} \leq |\!|\!|\theta|\!|\!|_{|s|, \varOmega_{-}} \leq \sqrt{\frac{|s|}{\underline{\sigma} } }|\!|\!|\theta|\!|\!|_{1,\varOmega_{-}}, \end{align} (4.5)  \begin{align} \underline{\sigma } |\!|\!|v|\!|\!|_{1, \varOmega_{+}} \leq |\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}\leq \frac{|s|}{\underline{\sigma}} |\!|\!|v|\!|\!|_{1, \varOmega_{+}}, \end{align} (4.6)which can be obtained from the inequalities   $$ \min\{1, \sigma\} \leq \min\{1, |s|\}\quad \textrm{and} \quad \max\{1, |s|\} \min\{1, \sigma\} \leq |s|, \quad \forall\, s \in \mathbb{C}_{+}. $$We remark that the norms $$|\!|\!|\theta |\!|\!|_{1, \varOmega _{-}}$$ and $$ |\!|\!| v|\!|\!|_{1, \varOmega _{+}}$$ are equivalent to $$\| \theta \|_{H^{1}(\varOmega _{-})} $$ and $$ \|{ v}\|_{H^{1}(\varOmega _{+})}$$, respectively, and also the energy norm $$|\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega _{-}}$$ is equivalent to the $$\mathbf{H}^{1}(\varOmega _{-})$$ norm of u by the second Korn inequality (Fichera, 1973). For the invertibility of $$\pmb{\mathcal{A} }$$, consider two invertible (for $$s\in \mathbb C_{+}$$) diagonal matrices   $$ \pmb{\mathcal Z_{1}}:= \left(\begin{array}{cccccc} \rho_{f}^{-1} & 0 & 0 & 0 \\ 0 & \eta^{-1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & s \end{array}\right), \qquad \pmb{\mathcal Z_{2}}:= \left(\begin{array}{cccccc} s^{-1} & 0 & 0 & 0 \\ 0 & \zeta^{-1}\rho_{f} & 0 & 0 \\ 0 & 0 & s^{-1} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$and note that   $$ \pmb{\mathcal Z_{1}}\pmb{\mathcal{A}}\pmb{\mathcal Z_{2}} =\pmb{\mathcal{B}}:=\left(\begin{array}{cccccc} (\rho_{f}\,s)^{-1}\mathbf A_{s}& -\textrm{div}^{\prime}& s \gamma_{n}^{\prime} & 0 \\ \textrm{div} & \frac{\rho_{f}}{\eta\zeta} B_{s} &0&0 \\ -\gamma_{n} & 0 & s^{-1} W(s) & - \frac{1}{2} I + K^{\prime}(s) \\ 0 & 0 & \frac{1}{2}I - K(s) &s V(s) \end{array}\right). $$By Banjai et al. (2015, Lemma 3.1), the block   $$ \left(\begin{array}{cccccc} s^{-1} W(s) & - \frac{1}{2} I + K^{\prime}(s) \\ \frac{1}{2}I - K(s) &s V(s) \end{array}\right) $$is coercive and it is therefore very simple to prove  $$ \textrm{Re}\Big( \big\langle \pmb{\mathcal{B}} (\mathbf v, \vartheta, \psi, \chi), \overline{ (\mathbf v, \vartheta, \psi, \chi) } \big\rangle \Big) \geq C_{1}\big(\zeta, \eta, \rho_{f} \big) \; C_{2}\big(|s|,\textrm{Re}\,s\big)\big \| (\mathbf v, \vartheta, \psi, \chi)\big\|^{2}_{X} $$for some positive $$C_{1}$$ and $$C_{2}$$ that are not relevant for this argument. This proves that $$\pmb{\mathcal{A}}$$ is invertible. Let us next study solutions of the problem   \begin{align} \pmb{\mathcal{A}} \left(\begin{array}{cccccc} \mathbf u \\ \theta \\ \phi \\ \lambda\end{array}\right) = \left(\begin{array}{cccccc} d_{1} \\ 0 \\ d_{3} \\ 0 \end{array}\right). \end{align} (4.7)Note that we have taken $$d_{2}=0$$ and $$d_{4}=0$$ as we did in the first occurrence of the operator $$\pmb{\mathcal{A}}$$ in (3.13). If we define   \begin{align} v:= \mathcal{D}(s) \phi - \mathcal{S}(s) \lambda \quad \textrm{in} \quad \mathbb{R}^{3} \setminus \varGamma, \end{align} (4.8)then $$v\in H^{1}(\mathbb{R}^{3} \setminus \varGamma )$$ is the solution of the transmission problem   \begin{align} - \varDelta v + (s/c)^{2} v = 0 \quad \textrm{in}\quad \mathbb{R}^{3} \setminus \varGamma \end{align} (4.9)satisfying the following jump relations across $$\varGamma $$:   $$ {[\kern-1.5pt[}\gamma v{]\kern-1.5pt]}:= \gamma^{+} v - \gamma^{-} v = \phi \in H^{1/2}\big(\varGamma\big), \quad {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} :=\partial_{n}^{+} v - \partial_{n}^{-} v = \lambda \in H^{-1/2}\big(\varGamma\big). $$The last two equations from (4.7) are then equivalent to   \begin{align} -s \;\gamma^{-}{\mathbf u }\cdot\mathbf n - \partial_{n}^{+} v = d_{3}\quad \textrm{on} \quad \varGamma, \end{align} (4.10)  \begin{align} - \; \gamma^{-} v = 0\quad \textrm{on} \quad \varGamma. \end{align} (4.11)The boundary conditions (4.11) and (4.9) imply that v = 0 in $$\varOmega _{-}$$ and therefore   \begin{align} {[\kern-1.5pt[}\gamma v{]\kern-1.5pt]} = \gamma^{+} v = \phi \quad \textrm{and} \quad {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} = \partial_{n}^{+} v = \lambda. \end{align} (4.12)We introduce the bilinear form in $$H^{1}(\varOmega _{+})$$,  $$ c_{\varOmega_{+}} \big(f, g; s\big):= \big(\nabla f,\nabla g\big)_{\varOmega_{+}} + \big(s/c\big)^{2}\big(f,g\big)_{\varOmega_{+}}, $$and the associated operator   $$\begin{aligned} C_{s, \varOmega_{+}}:&& \; H^{1}(\varOmega_{+}) \;& \longrightarrow\;& \left(H^{1}(\varOmega_{+})\right)^{\prime}\!,\\ &&\; f \;& \longmapsto \;& c_{\varOmega_{+}}(f,\;\cdot\;;s), \end{aligned}$$ By (4.9) and the transmission condition (4.10), we obtain   \begin{align} \big(C_{s, \varOmega_{+} }\, v,w\big)_{\varOmega_{+}} = &- \big\langle \partial^{+}_{n} v, \gamma^{+} w\big \rangle \nonumber\\ \,= &\, \big\langle d_{3}, \gamma^{+} w\big\rangle_{\varGamma} + \big\langle s ~\gamma^{-} \mathbf{u} \cdot \; \mathbf{n}, \gamma^{+} w\big \rangle_{\varGamma}. \end{align} (4.13)Therefore, the triple   $$ (\mathbf{u}, \theta, v) \in \pmb{\mathbb{H}} = {\mathbf H}^{1}(\varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^ 1(\varOmega_{+}) $$satisfies the system of operator equations   \begin{align} \left(\begin{array}{cccccc} \mathbf A_{s} & -\zeta \,\textrm{div}^{\prime} & s \rho_{f} \gamma_{n}^{\prime} \gamma^{+} \\ s\,\eta\,\textrm{div} & B_{s} & 0 \\ -s(\gamma^{+})^{\prime} \gamma_{n} & - & C_{s,\varOmega_{+}} \end{array}\right) \left(\begin{array}{cccccc} \mathbf u \\ \theta \\ v \end{array}\right) = \left(\begin{array}{cccccc} d_{1} \\ 0 \\ (\gamma^{+})^{\prime}d_{3} \end{array}\right). \end{align} (4.14)Returning to bilinear forms, these equations are equivalent to   \begin{align} \begin{aligned} a(\mathbf{u}, \mathbf v;s ) \! \! - \! \zeta( \theta,{\nabla\cdot} \; \mathbf v )_{\varOmega_{-}}\! \! + \!s \rho_{f} \langle \gamma^{+} v , \gamma^{-} \mathbf v \cdot\mathbf n\rangle_{\varGamma} = \,& (d_{1}, \mathbf v )_{\varOmega_{-}} \quad && \forall\, \mathbf v \in{\mathbf H}^{1}(\varOmega_{-}), \\ s\,\eta\,(\nabla\cdot \; \mathbf u, {\vartheta } )_{\varOmega_{-}} + b(\theta, \; \vartheta;s ) =\;& 0 \quad&& \forall \, \vartheta \in H^{1}(\varOmega_{-}), \\ - s \langle \gamma^{-} \mathbf{u} \cdot\mathbf n\;, \; {\gamma^{+} v} \rangle_{\varGamma}+ c_{\varOmega_{+}}( v,w;s) =\; & \langle d_{3}, {\gamma^{+} w} \rangle_{\varGamma} \quad&& \forall \ w \in H^{1}(\varOmega_{+}). \end{aligned} \end{align} (4.15)Note that the expression $$(d_{1}, \mathbf v )_{\varOmega _{-}}$$ in the right-hand side of (4.15) is a duality product and not an $$L^{2}$$ inner product. By construction, the variational problem (4.15) is equivalent to the operator equations (4.7) when we identify on the one hand $$\phi =\gamma ^{+} v$$ and $$\lambda =\partial _{\nu }^{+} v$$ and, on the other hand, $$v=\mathcal D(s)\phi -\mathcal S(s)\lambda $$. Theorem 4.1 The variational problem (4.15) has a unique solution $$(\mathbf{u}, \theta , v) \in \pmb{\mathbb{H}} $$. Moreover, the following estimate holds:   \begin{align} \left(\!|\!|\!|\mathbf{u}|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!|s^{-1/2}\theta|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!|v|\!|\!|^{2}_{|s|, \varOmega_{+}} \!\right)^{1/2} \leq c_{0} \; \frac{|s|}{\sigma{\underline{\sigma}}} ~ \big\|(d_{1}, 0, d_{3}, 0)\big\|_{X^{\prime}}, \end{align} (4.16)where $$c_{0}$$ is a constant depending only on the physical parameters $$ \rho _{f}, \zeta , \eta $$. Proof. Testing equations (4.15) with $$\mathbf v=\overline{s\,\mathbf u}$$, $$\vartheta =\zeta \,\eta ^{-1}\overline \theta $$, $$v=\rho _{f}\,\overline{s\, v}$$, and adding the result, it is simple to see that   \begin{align} \textrm{Re}\,\Big( \overline s\, &a(\mathbf u,\overline{\mathbf u};s) +\zeta\,\eta^{-1}\, b(\theta,\overline\theta;s) +\rho_{f} \overline s\,c_{\varOmega_{+}} (v,\overline v;s)\Big) \\ \nonumber &=\textrm{Re}\, \Big( \overline s \, (d_{1},\overline{\mathbf u})_{\varOmega_{-}} +\overline s\,\langle d_{3},\gamma^{+}\overline v\rangle_{\varGamma}\Big). \end{align} (4.17)A simple computation shows next that   \begin{align} \textrm{Re} \left( \overline s a(\mathbf u, \overline{\mathbf u};s)\right) =\; \sigma\, |\!|\!|\mathbf u|\!|\!|^{2}_{|s|,\varOmega_{-}}, \end{align} (4.18)  \begin{align} \textrm{Re}\, b(\theta,\overline{\theta};s) \geq\; \frac{\sigma}{|s|}~ |\!|\!|\theta|\!|\!|^{2}_{|s|, \varOmega_{-}}=\sigma |\!|\!|s^{-1/2}\theta|\!|\!|_{|s|,\varOmega_{-}}^{2}, \end{align} (4.19)  \begin{align} \textrm{Re} \left( \overline s\, c_{\varOmega_{+}} (v, \overline{v};s) \right) =\;\sigma\, |\!|\!|v|\!|\!|^{2}_{|s|,\varOmega_{+}}, \end{align} (4.20)which proves the well-posedness of the variational problem (4.15). Using these inequalities in (4.17) for the particular case $$d_{2}=0$$, and taking into account the inequalities (4.4) and (4.6) (relating energy norms to Sobolev norms), we can easily prove the result. Theorem 4.2 If   $$ (d_{1},d_{3})\in (\mathbf H^{1}(\varOmega_{-}))^{\prime}\times H^{-1/2}(\varGamma), $$then for the unique solution of (4.7) we have the estimates   \begin{align} |\!|\!|\mathbf u|\!|\!|_{1,\varOmega_{-}}+\big\|\phi\big\|_{H^{1/2}(\varGamma)} \leq c_{1} \frac{|s|}{\sigma\,\underline\sigma^{2}} \big\| \big(d_{1},0,d_{3},0\big)\big\|_{X^{\prime}}, \end{align} (4.21)  \begin{align} |\!|\!|\theta|\!|\!|_{1,\varOmega_{-}}+\big\|\lambda\big\|_{H^{-1/2}(\varGamma)} \leq c_{2} \frac{|s|^{3/2}}{\sigma\,\underline\sigma^{3/2}} \big\| \big(d_{1},0,d_{3},0\big)\big\|_{X^{\prime}}, \end{align} (4.22)where $$c_{1}$$ and $$c_{2}$$ are constants independent of s. Proof. The bounds for u and $$\theta $$ follow from Theorem 4.1 and (4.4)–(4.5). Since, as we pointed out in (4.12), we have   $$ \gamma^{+}v={[\kern-1.5pt[}\gamma v{]\kern-1.5pt]} = \phi \in H^{1/2}(\varGamma), \quad \partial_{n}^{+} v= {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} =\lambda \in H^{-1/2}(\varGamma), $$we thus have the estimate (see, e.g., Hsiao et al., 2013)   \begin{align} \big\|\phi\big \|_{H^{1/2}(\varGamma)} = \big\|\gamma^{+} v\big\|_{H^{1/2}(\varGamma)} \leq C |\!|\!|v|\!|\!|_{1, \varOmega_{+}} \leq C \frac{1}{{\underline{\sigma}}} |\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}, \end{align} (4.23)therefore the bound for $$\phi $$ follows from Theorem 4.1 again. Similarly, an application of Bamberger & Ha-Duong’s (1986) optimal lifting leads to the estimate   \begin{align} \big\|\lambda\big\|_{H^{-1/2}(\varGamma)} =\big\|\partial^{+}_{n} v\big \|_{H^{-1/2}(\varGamma)} \leq c_{2} \big(|s|/\underline\sigma\big)^{1/2}|\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}. \end{align} (4.24)A detailed proof of this inequality can be found in Sayas (2016, Proposition 2.5.2). Therefore, Theorem 4.1 provides the bound for $$\lambda $$. We remark that in view of (3.2), we see that $$\mathbf{u}, \theta $$ and v are solutions of the system   \begin{align} \left(\begin{array}{c} \mathbf{u} \\ \theta\\ v\\ \end{array}\right) = \left ( \begin{array}{cccc} 1& 0 & 0 & 0\\ 0 & 1& 0 & 0 \\ 0 & 0 &\mathcal{D}(s) & - \mathcal{S}(s)\\ \end{array} \right ) {\pmb{\mathcal{A}}}^{-1} \left(\begin{array}{c} d_{1} \\ 0\\ d_{3}\\ 0\\ \end{array}\right). \end{align} (4.25)With the properties of solutions available in the transformed domains, we are now in a position to estimate the corresponding properties of solutions in the time domain based on some inversion theorems of the Laplace transform (see Lubich, 1994 for an early version and Sayas, 2016, Proposition 3.2.2 for the result as we use it). The crucial result described in Proposition 4.3 below is employed to retrieve time-domain estimates from those obtained in the Laplace domain. Before presenting the aforementioned result we must introduce some notation. For Banach spaces X and Y, let $$\mathcal{B}(X, Y)$$ denote the set of bounded linear operators from X to Y. We say that an analytic function $$A : \mathbb{C}_{+} \rightarrow \mathcal{B}(X, Y)$$ is an element of the class of symbols $$\mathbf{Sym} (\mu , \mathcal{B}(X, Y))$$ if there exist $$\mu \in \mathbb{R}$$ and $$m\geq 0$$ such that   $$ \big\|A(s)\big\|_{X,Y} \leq C_{A}(\textrm{Re}~s) |s|^{\mu} \quad \textrm{for}\quad s \in \mathbb{C}_{+}, $$where $$C_{A} : (0, \infty ) \rightarrow (0, \infty ) $$ is a nonincreasing function such that   $$ C_{A}(\sigma) \leq \frac{ c}{\sigma^{m}} \quad \forall \, \sigma \in ( 0, 1].$$In order to make the statement of the time-domain estimates more compact, we will make use of the regularity spaces   \begin{align} W^{k}_{+}( \mathcal{H}):= \Big\{ w \in \mathcal{C}^{k-1}(\mathbb{R}; \mathcal{H}) : w ~\equiv 0~ \textrm{in} ~(-\infty,0), w^{(k)} \in L^{1} (\mathbb{R}; \mathcal{H})\Big \}, \end{align} (4.26)where $$\mathcal{H}$$ denotes a Banach space. Proposition 4.3 (Laliena & Sayas, 2009; Sayas, 2016). Let $$A = \mathcal{L}\{a\} \in \mathbf{Sym}\big (\mu , \mathcal{B}\big (X, Y\big )\big )$$ with $$\mu \geq 0$$ and let   $$ k:=\big\lfloor \mu +2\big \rfloor, \quad \varepsilon := k - (\mu +1) \in (0, 1]. $$If $$ g \in W_{+}^{k}(\mathbb R,X)$$, then $$a* g \in \mathcal{C}(\mathbb{R}, Y)$$ is causal and   $$ \big\|\big (a*g\big)\big(t\big)\big \|_{Y} \leq 2^{\mu+1} C_{\varepsilon} \big(t\big) C_{A} \big(t^{-1}\big) {\int_{0}^{1}} \big\|\big(\mathcal{P}_{k}g\big)\big(\tau\big)\big \|_{X} \; \textrm{d}\tau, $$where   $$ C_{\varepsilon} \big(t\big) := \frac{t^{\varepsilon} }{\pi \varepsilon } \qquad\textrm{and}\qquad \big(\mathcal{P}_{k}g\big) \big(t\big) = \sum_{\ell=0}^{k} {{k}\choose{\ell}} g^{(\ell)}(t). $$ Theorem 4.4 If $$\mathbf{D} := \mathcal{L}^{-1}\{ (d_{1}, 0, d_{3}, 0)\}\in W_{+}^{3}(\mathbb R, X^{\prime })$$, then   \begin{align*} \big(\mathbf{U}, \varTheta, \mathcal{L}^{-1} \big\{\phi\big\}, \mathcal{L}^{-1}\big\{ \lambda\big \}\big) & \in \mathcal{C}\big(\big[0,T\big],X\big),\\ \big(\mathbf{U}, \varTheta, V\big) \in C\big( \big[0, T\big], \pmb{\mathbb{H}} \big) \end{align*}and we have the bounds   \begin{align*} \big\| \mathbf U(t)\big\|_{\mathbf H^{1}(\varOmega_{-})} + \big\|\mathcal L^{-1}\big\{\phi\big\}(t)\big\|_{H^{1/2}(\varGamma)} +\big\| V(t)\big\|_{H^{1}(\varOmega_{+})} & \leq c t^{2} \max\big\{1,t^{2}\big\} {\int_{0}^{t}} \big\| (\mathcal{P}_{3}\mathbf D) (\tau)\big\|_{X^{\prime}}\; \textrm{d} \tau,\\ \big\| \varTheta(t)\big\|_{H^{1}(\varOmega_{-})} + \big\|\mathcal L^{-1}\big\{\lambda\big\}\big\|_{H^{-1/2}(\varGamma)} &\leq c t^{\frac32} \max\big\{1,t^{\frac32}\big\} {\int_{0}^{t}} \big\| (\mathcal{P}_{3}\mathbf D) (\tau)\big\|_{X^{\prime}}\; \textrm{d} \tau. \end{align*} Proof. For the bounds for U(t) and $$\mathcal L^{-1}\big \{\phi \big \}(t)$$, use (4.21) and Proposition 4.3 with $$\mu =1$$, k = 3, $$\varepsilon =1$$. Note that a bound for V (t) follows from the same proposition now using Theorem 4.1 and (4.6). The bounds for $$\varTheta (t)$$ and $$\mathcal L^{-1}\big \{\lambda \big \}(t)$$ follow from (4.22) and Proposition 4.3 with $$\mu =3/2$$, k = 3, $$\varepsilon =1/2$$. 5. Semidiscrete error estimates In this section we discuss the results concerning the discretization of (3.13). We begin with the Galerkin semidiscretization in the space of the system of equations. Let   $$ {\mathbf V}_{h} \subset \mathbf H^{1}(\varOmega_{-}), \quad W_{h} \subset H^{1}(\varOmega_{-}), \quad Y_{h} \subset H^{1/2}(\varGamma), \quad X_{h} \subset H^{-1/2} (\varGamma) $$be families of finite-dimensional subspaces. We say $$ \left (\mathbf u^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\right ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$ is a Galerkin solution of (4.7) if it satisfies the Galerkin equations   \begin{align} \Big(\pmb{\mathcal{A}} \big(\mathbf u^{h}, \theta^{h}, \phi^{h}, \lambda^{h}\big)^{\textrm{T}},\big(\mathbf v,\vartheta,\psi,\eta\big)^{\textrm{T}}\Big) = \Big(\big(d_{1},d_{2},d_{3},0\big)^{\textrm{T}},\big(\mathbf v,\vartheta,\psi,\eta\big)^{\textrm{T}}\Big) \end{align} (5.1)for all $$(\mathbf v, \vartheta , \psi , \eta ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$. Solutions of Galerkin equations of (5.1) can be established in the same manner as the exact solutions of the system (4.7). We will not repeat the process and we consider only the error estimates here. We note that if $$\big ({\mathbf u}^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\big ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$ is a Galerkin solution of (5.1), then   \begin{align} v^{\,h} := \mathcal{D}(s) \phi^{h} - \mathcal{S}(s) \lambda^{h} \; \in H^{1}\big(\mathbb R^{3}\setminus\varGamma\big) \end{align} (5.2)satisfies   \begin{align} -\varDelta v^{h} + (s/c)^{2}v^{h} = \; & 0 \qquad \mbox{in } \quad \mathbb{R}^{3} \setminus \varGamma,\\ \nonumber {[\kern-1.5pt[}\gamma v^{h}{]\kern-1.5pt]} = \;& \phi^{h}\in Y_{h} \subset H^{1/2}(\varGamma), \\ \nonumber {[\kern-1.5pt[}\partial v^{h}{]\kern-1.5pt]} = \;& \lambda^{h} \in X_{h} \subset H^{-1/2}(\varGamma). \end{align} (5.3)Now set   $$ V_{h} := \big\{ w \in H^{1}\big(\mathbb{R}^{3} \setminus \varGamma \big) : {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]} \in Y_{h}, \gamma^{-}w \in X^{\circ}_{h} \big\}. $$Here and in the sequel, the upper script $$^{\circ }$$ will be used to denote the annihilator (or polar set) of a subspace of a Banach space. In this particular case,   $$ X_{h}^{\circ} := \big\{ w \in X^{\prime}_{h} : \big\langle v, w\big \rangle = 0 \quad \forall\, v \in X_{h} \big\}. $$Applying Green’s formula to (5.3), we obtain for $$w \in V_{h}$$,   \begin{align} \big\langle \partial_{n}^{+} v^{h}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} = - c_{\mathbb R^{3}\setminus\varGamma}\;\big(v^{h}, w\; ; s\big) - \big\langle{[\kern-1.5pt[}\partial_{n} v^{h}{]\kern-1.5pt]},\gamma^{-} w\big\rangle_{\varGamma}. \end{align} (5.4)As a consequence, we see that $$ \big ( \mathbf u^{h}, \theta ^{h}, v^{h}\big ) \in{\mathbf V}_{h} \times W_{h} \times V_{h} $$ satisfies the variational equations   \begin{align} a\big(\mathbf{u}^{h}, \mathbf v; s \big) - \zeta \big( \theta^{h}, \nabla\cdot \; \mathbf v \big)_{\varOmega_{-}}\! + s\rho_{f}\big\langle {[\kern-1.5pt[}\gamma v^{h}{]\kern-1.5pt]}, \gamma^{-} \mathbf v \cdot \mathbf n\big\rangle_{\varGamma} = \; & \big(d_{1}, \mathbf v\big)_{\varOmega_{-}} &\quad& \forall\, \mathbf v \in{\mathbf V}^{h}, \nonumber \\ s\, \eta\, \big( \nabla\cdot\; \mathbf u^{h}, \vartheta\big)_{\varOmega_{-}}\! + b\big(\theta^{h}, \vartheta; s\big) =\;& \big(d_{2}, \vartheta\big)_{\varOmega_{-}}&\quad& \forall\, \vartheta \in W_{h}, \\ - s\big \langle \gamma^{-} \mathbf{u}^{h} \cdot \mathbf{n}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} + c_{\mathbb R^{3}\setminus \varGamma} \big( v^{h}, w; s\big ) =\;& \big\langle d_{3}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} &\quad& \forall\ w ~ \in V_{h}. \nonumber \end{align} (5.5)For the error estimate, we need to compare $$\big (\mathbf u^{h}, \theta ^{h}, v^{h}\big )$$ with the exact solution $$\big (\mathbf u, \theta , v\big )$$ of the transmission problem. The exact solution $$\big (\mathbf u, \theta , v\big ) \in \mathbf H^{1}\big (\varOmega _{-}\big ) \times H^{1}\big (\varOmega _{-}\big ) \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big )$$ satisfies the variational equation (4.15), understanding that v ≡ 0 in $$\varOmega _{-}$$. This implies that   \begin{align} a\big(\mathbf{u}^{h}\!- \mathbf u, \mathbf v; s \big) - \zeta\big( \theta^{h} \!- \theta, \nabla\cdot \; \mathbf v \big)_{\varOmega_{-}} \! + s\; \rho_{f} \big\langle[\![\gamma \big(v^{h} \!-v\big)]\!] , \gamma^{-} \mathbf v\cdot \mathbf n \big\rangle_{\varGamma} &\; = 0 \quad \forall\, \mathbf v \in{\mathbf V}^{h}, \nonumber \\ \frac{s}{|s|} \zeta\left(\nabla\cdot\big(\mathbf u^{h} \!- \mathbf u\big), \vartheta \right)_{\varOmega_{-}}\! + \frac{\zeta}{\eta} \frac{1}{|s|} b\big(\theta^{h}\!- \theta, \vartheta; s\big) &\; = 0 \quad \forall\, \vartheta \in W_{h}, \\ \nonumber - s\rho_{f} \big\langle \gamma^{-} \big(\mathbf{u}^{h}\!- \mathbf u\big)\cdot\mathbf n, [\![\gamma w]\!]\big\rangle_{\varGamma} + \rho_{f} c_{\mathbb R^{3}\setminus \varGamma} \big( v^{h}\! - v, w; s \big) \quad & & & \\ \nonumber + \rho_{f} \big\langle [\![\partial_{n}\big(v^{h}\! - v\big)]\!], \gamma^{-} w \big\rangle_{\varGamma} &\; = 0 \quad \forall\ w \in V_{h}. \nonumber \end{align} (5.6)Note that we have multiplied the second and third equations by $$\eta /(|s|\zeta )$$ and $$\rho _{f}$$, respectively, mimicking what we did in the proof of Theorem 4.1. The significance of (5.6) is that it indicates that the Galerkin solutions are the best possible approximations of the exact solution in the finite-dimensional subspaces with respect to the inner products defined by the underlying bilinear forms. However, it is worth emphasizing that the errors $$ \big (\mathbf u^{h} -\mathbf u \big )$$ and $$\big (\theta ^{h} - \theta \big )$$ do not belong to the discrete function spaces. In order to justify (5.6) as a proper variational formulation, we will make use of the spaces of constants and of infinitesimal rigid motions   \begin{align} \mathfrak{R}_{\mathbf u} := \;\big\{ \mathbf m \in \mathbf H^{1}(\varOmega_{-}): ( \mathbf C \boldsymbol\varepsilon (\mathbf m), \boldsymbol{\varepsilon}(\mathbf m))_{\varOmega_{-}} = 0 \big\}, \end{align} (5.7)  \begin{align} {\mathfrak{R}}_{\theta} := \; \big\{ m \in H^{1}(\varOmega_{-}): ( \nabla m, \nabla m)_{\varOmega_{-}} = 0 \big\}, \end{align} (5.8)which in what follows will always be assumed to be contained in the discrete subspaces $$\mathbf V_{h}$$ and $$W_{h}$$, respectively. We now define the elliptic projection on displacement fields   \begin{align} \begin{aligned} \mathbf P_{h}: \mathbf H^{1}(\varOmega_{-}) \longrightarrow \;&\mathbf V_{h} \subset \mathbf H^{1}(\varOmega_{-}), & \\ \big(\mathbf C\boldsymbol\varepsilon(\mathbf P_{h} \mathbf u), \boldsymbol{\varepsilon} \big(\mathbf v^{h}\big)\big)_{\varOmega_{-}} = \;& \big(\mathbf C\boldsymbol\varepsilon(\mathbf u), \boldsymbol{\varepsilon} \big(\mathbf v^{h} \big) \big)_{\varOmega_{-}} \qquad & \forall\, \mathbf v^{h}\in \mathbf V_{h},\\ (\mathbf P_{h} \mathbf u, \mathbf m)_{\varOmega_{-}} =\;& (\mathbf u, \mathbf m)_{\varOmega_{-}} \qquad & \forall\ \mathbf m\in \mathfrak R_{\mathbf u}. \end{aligned} \end{align} (5.9)This projection is well defined thanks to Korn’s second inequality and can be alternatively introduced as the orthogonal projection onto $$\mathbf V_{h}$$ with respect to the nonstandard (but equivalent) inner product in $$\mathbf H^{1}(\varOmega),$$  $$ (\mathbf C\boldsymbol\varepsilon(\mathbf u),\boldsymbol\varepsilon(\mathbf v))_{\varOmega_{-}}+\big(\mathbf P_{\mathfrak R_{\mathbf u}}\mathbf u,\mathbf P_{\mathfrak R_{\mathbf u}}\mathbf v\big)_{\varOmega_{-}}, $$where $$\mathbf P_{\mathfrak R_{\mathbf u}}$$ is the $$\mathbf L^{2}(\varOmega _{-})$$ orthogonal projection onto $$\mathfrak R_{\mathbf u}$$. This implies that the approximation error $$\|\mathbf u-\mathbf P_{h}\mathbf u\|_{1,\varOmega _{-}}$$ is equivalent to the best approximation error in $$\mathbf H^{1}(\varOmega _{-})$$ by elements of $$\mathbf V_{h}$$. Similarly, we can introduce a projection on the discrete scalar fields   \begin{align} \begin{aligned} Q_{h}: H^{1}(\varOmega_{-}) \longrightarrow \;& W_{h} \subset H^{1}(\varOmega_{-}), & \\ \big(\nabla (Q_{h} \theta), \nabla \vartheta^{h}\big)_{\varOmega_{-}} = \;& \big(\nabla \theta, \nabla\vartheta^{h} \big)_{\varOmega_{-}} \qquad & \forall\,\vartheta^{h}\in W_{h},\\ (Q_{h} \theta, m)_{\varOmega_{-}} = \;&(\theta, m)_{\varOmega_{-}}\qquad &\forall\ m\in \mathfrak R_{\theta}. \end{aligned} \end{align} (5.10)Note that the reason to introduce these projections is to avoid having additional mass terms arising from the full Sobolev norm in the associated error equations. In terms of the elliptic projection $$\mathbf P_{h}$$, we can define   $$ \mathbf e_{\mathbf u}^{h} := \mathbf u^{h} - \mathbf P_{h}\mathbf u, \qquad \mathbf r_{\mathbf u}^{h} := \mathbf P_{h} \mathbf u - \mathbf u, $$so that we can decompose   $$ \mathbf u^{h} - \mathbf u = \big(\mathbf u^{h} - \mathbf P_{h} \mathbf u \big) + (\mathbf P_{h} \mathbf u - \mathbf u) = \mathbf e_{\mathbf u}^{h} + \mathbf r_{\mathbf u}^{h}. $$As a consequence,   $$ a \big(\mathbf u^{h} - \mathbf u, \mathbf v^{h} ; s\big) = a \big(\mathbf e_{\mathbf u}^{h} \;, \mathbf v^{h}; s\big) + s^{2} \rho_{\varSigma} \big(\mathbf r_{\mathbf u}^{h}, \mathbf v^{h}\big)_{\varOmega_{-}}. $$We may decompose the error $$\theta ^{h} - \theta $$ in a similar manner by letting   $$ e^{h}_{\theta}:= \theta^{h} - Q_{h}\theta, \qquad r^{h}_{\theta} := Q_{h}\theta - \theta, $$so that $$\theta ^{h}-\theta = e^{h}_{\theta } + r^{h}_{\theta }$$ and therefore   $$ b \big(\theta^{h} -\theta, \vartheta^{h}; s \big) = b \big(e^{h}_{\theta}, \vartheta^{h}; s\big) + (s/\kappa) (r^{h}_{\theta}, \vartheta^{h})_{\varOmega_{-}}. $$Finally, we define   $$ {e_{v}^{h}}:= \mathcal D(s) (\phi^{h}-\phi) - \mathcal S(s)(\lambda^{h}-\lambda) \qquad \textrm{ in }\; \mathbb R^{3}\setminus\varGamma. $$This leads to the variational formulation for the error functions $$\big (\mathbf e_{\mathbf u}^{h}, e^{h}_{\theta }, {e^{h}_{v}}\big ) \in \mathbf V_{h} \times W_{h} \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big )$$. Theorem 5.1 The error functions $$ \big (\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta }, {e^{h}_{v}}\big ) \in \mathbf V_{h} \times W_{h} \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big ) $$ satisfy the variational formulation   \begin{align*} \big(\gamma^{-} {e^{h}_{u}},\; {[\kern-1.5pt[}\gamma{e^{h}_{u}}{]\kern-1.5pt]}\; +\phi,\; {[\kern-1.5pt[}\partial_{n} {e^{h}_{v}}{]\kern-1.5pt]} + \lambda\big) \in \; & X^{\circ}_{h} \times Y_{h}\times X_{h}, & \\ \mathcal{A}((\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}), ( \mathbf v, \vartheta, w) ;s ) =\; & \ell((\mathbf v, \vartheta, w) ; s) \quad\forall\, (\mathbf v, \vartheta, w) \in \mathbf V_{h} \times W_{h} \times V_{h}. \end{align*}Where the bilinear form $$\mathcal A$$ is defined by   \begin{align} \mathcal{A}\big(\big(\mathbf{e}^{h}_{\mathbf{u}}, e^{h}_{\theta}, {e^{h}_{v}}\big), ( \mathbf{v}, \vartheta, w) ;s \big) :=\;& a \big(\mathbf{e}^{h}_{\mathbf{u}}, \mathbf{v}; s \big) + \frac{s}{|s|} \zeta\big({\nabla\cdot} ~ \mathbf{e}^{h}_{\mathbf{u}}, \vartheta\big)_{\varOmega_{-}} - s\rho_{f}\big\langle \gamma^{-} \mathbf{e}^{h}_{\mathbf{u}} \;, [\![\gamma w]\!]\mathbf{n}\big\rangle_{\varGamma} \nonumber \\ & + \frac{\zeta}{\eta} \frac{1}{|s|} b\; \big(e^{h}_{\theta}, \vartheta; s\big) - \zeta \big(e^{h}_{ \theta}, {\nabla\cdot}\; \mathbf{v}\big)_{\varOmega_{-}} \nonumber \\ & +\rho_{f} c_{\mathbb{R}^{3}\setminus \varGamma} \; \big( {e^{h}_{v}}, w; s \big) + s \rho_{f}\big\langle [\![\gamma{e^{h}_{v}}]\!] \; \mathbf{n}, \gamma^{-} \mathbf{v} \big\rangle_{\varGamma}, \end{align} (5.11)and the functional ℓ is given by   \begin{align} \nonumber \ell ((\mathbf v, \vartheta, w ) ; s) :=\; & - s^{2} \rho_{\varSigma}\big(\mathbf r^{h}_{\mathbf u}, \mathbf v \big)_{\varOmega_{-}} + \zeta\big(r^{h}_{\theta},{\nabla\cdot}~\mathbf v\big)_{\varOmega_{-}} -\frac{s}{|s|} \zeta \big({\nabla\cdot}~\mathbf r^{h}_{\mathbf u}, \vartheta\big)_{\varOmega_{-}}\\ & -\frac{\zeta}{\eta} \frac{s}{|s|}\frac{1}{\kappa}\big( r^{h}_{\theta}, \vartheta\big)_{\varOmega_{-}} +s \rho_{f} \big\langle \gamma^{-} \mathbf r^{h}_{ \mathbf u}, [\![\gamma w]\!] \mathbf n \big\rangle_{\varGamma} + \rho_{f} \langle \lambda, \gamma^{-}w \rangle_{\varGamma}. \end{align} (5.12) Proof. The bilinear form $$\mathcal{A}$$ follows easily from the left-hand side of (5.5) replacing $$\big (\mathbf{u}^{h} - \mathbf{u}, \theta _{h} - \theta , v^{h} - v\big ) $$ by $$\big ( \mathbf{e}^{h}_{\mathbf{u}} +\mathbf{r}^{h}_{\mathbf{u}}, e^{h}_{\theta } + r^{h}_{\theta }, {e^{h}_{v}}\big ) $$ and taking special care of the term $$e^{h}_{\theta }$$. From Green’s formula (5.4), we have   $$ \big\langle \partial_{n}^{+} {e^{h}_{v}}\;, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle = - c_{\mathbb R^{3}\setminus \varGamma}\;\big({e^{h}_{v}}, w\; ; s\big) - \big\langle{[\kern-1.5pt[}\partial_{n} {e_{v}^{h}}{]\kern-1.5pt]}, \gamma^{-} w~ \big\rangle. $$But equations (4.10) and (4.11) imply   $$ -s \mathbf n\cdot \big(e^{h}_{\mathbf u} + \mathbf r^{h}_{\mathbf u}\big) -\partial_{n}^{+} {e^{h}_{u}} \in Y_{h}^{\circ} \quad \textrm{and} \quad \gamma^{-} e^{h} \in X^{\circ}_{h}. $$Hence,   $$ - s\rho_{f} \big\langle \gamma^{-} \mathbf e^{h}_{\mathbf u}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\mathbf{n} \big\rangle_{\varGamma} + \rho_{f} c_{\mathbb R^{3}\setminus \varGamma} \; \big( {e^{h}_{v}}, w; s \big)= s \rho_{f}\big\langle \gamma^{-} \mathbf r^{h}_{ \mathbf u}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]} \mathbf n\big\rangle_{\varGamma} -\rho_{f}\big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}, \gamma^{-} w \big\rangle_{\varGamma}. $$We can rewrite the last term on the right-hand side as   $$ -\rho_{f} \big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}, \gamma^{-} w \big\rangle_{\varGamma}= -\rho_{f} \big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}+ \lambda - \lambda, \gamma^{-} w \big\rangle_{\varGamma} = \rho_{f} \langle \lambda, \gamma^{-} w \rangle_{\varGamma}, $$where we have used that $$\lambda ^{h} = [\![\partial _{n} {e^{h}_{v}}]\!] + \lambda \in X_{h} $$, and $$ \gamma ^{-} w \in X^{\circ }_{h} $$ but $$ \lambda \not \in X_{h}$$. This completes the proof. Following arguments similar to those employed in the proof of Theorem 4.1, we can obtain the error estimate. In the following, for simplicity, let   $$ |\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}}\big)|\!|\!|^{2}_{|s|} := |\!|\!| \mathbf e^{h}_{\mathbf u}|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!| e^{h}_{\theta} |\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!| {e^{h}_{v}} |\!|\!|^{2}_{|s|, \mathbb{R}^{3} \setminus \varGamma} . $$ Theorem 5.2 For $$(\mathbf u, \theta , \phi , \lambda ) \in \mathbf H^{1}(\varOmega _{-}) \times H^{1}(\varOmega _{-}) \times H^{1/2}(\varGamma ) \times H^{-1/2} (\varGamma ) $$, there holds the error estimate   \begin{align} |\!|\!|\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)|\!|\!|_{|s|} \leq C \frac{|s|^{2}}{\sigma{\underline{\sigma}}^{3}} \Big(\| \lambda \|_{-1/2,\; \varGamma}^{2} + \|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\Big), \end{align} (5.13)where the constant C depends only on the geometry and physical parameters. Proof. It is easy to see from the definition of the bilinear form $$\mathcal{A}$$ in (5.11) that there is a constant C depending only on the geometry and physical parameters such that   \begin{align} |\mathcal{A} \big(\big(e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big), (\mathbf v, \vartheta, w); s\big)| \leq C \frac{|s|}{\sigma \underline{\sigma}} |\!|\!|\big(e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)|\!|\!|_{1}~ |\!|\!|(\mathbf v, \vartheta, w)|\!|\!|_{|s|} . \end{align} (5.14)We also need the estimate for the functional,   \begin{align} |\ell((\mathbf v, \vartheta, w);s)| \leq \frac{C}{\underline{\sigma}} \Big(\| \lambda \|_{-1/2,\; \varGamma} + \|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}}+ \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\Big)|\!|\!|(\mathbf v, \vartheta, w)|\!|\!|_{|s|}. \end{align} (5.15)For $$\phi \in H^{1/2} (\varGamma )$$, we pick a lifting $$v_{\phi } \in H^{1}(\mathbb{R}^{3} \setminus \varGamma )$$ such that $$\gamma ^{+} v_{\phi } = \phi , \gamma ^{-}v_{\phi } = 0$$. Thus,   $$ \| v_{\phi} \|_{1, \mathbb{R}^{3} \setminus \varGamma} \le C \| \phi \|_{1/2, \varGamma}. $$Since $$\big (\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta }, {e^{h}_{v}} + v_{\phi } \big ) \in \mathbf V_{h} \times W_{h} \times V_{h}$$, it follows from equations (4.18)–(4.20) that   \begin{align*} \frac{\sigma \underline{\sigma} }{|s|^{2}}|\!|\!|\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big)|\!|\!|^{2}_{|s|} \leq \;& \big |\mathcal{A}\big(\big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big), \big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big); s\big) \big |\\ =\; & \big | \ell \big( \big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big) ; s\big) + \mathcal{A} \big((\mathbf 0, 0, v_{\phi}), \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big) ;s ) \big | \\ \leq\; & \frac{C}{\underline\sigma}|\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}}+v_{\phi}\big)|\!|\!|_{|s|} \Big(\|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}}\!\!\\ &\;\; \qquad \qquad \qquad \qquad\qquad + \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\! +\| \lambda \|_{-1/2,\; \varGamma} + \!\frac{|s|}{\sigma} \| v_{\phi} \|_{\mathbb{R}^{3} \setminus \varGamma} \Big) \\ \leq\;& \frac{C}{\underline{\sigma}^{2}}|\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}} + v_{\phi}\big)|\!|\!|_{|s|} \Big(\|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} \\ &\;\; \qquad \qquad \qquad \qquad\qquad\ +\|r^{h}_{\theta}\|_{1, \varOmega_{-}} + \| \lambda \|_{-1/2,\; \varGamma} + \|s \phi\|_{1/2, \varGamma}\Big), \end{align*}and the result follows from this relation and the observation that   $$ |\!|\!|(\mathbf 0,0,v_{\phi})|\!|\!|\leq \frac{C}{\underline\sigma}\|s\phi\|_{1/2,\varGamma} .$$ We are now in a position to establish the following result. Corollary 5.3 If $$(\mathbf u, \theta , \phi , \lambda ) \in \mathbf H^{1}(\varOmega _{-}) \times H^{1}(\varOmega _{-}) \times H^{1/2}(\varGamma ) \times H^{-1/2}(\varGamma ) $$ is the unique solution of the problem (4.7) and $$\big (\mathbf u^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\big ) $$ is its Galerkin approximation (5.1), then we have the estimates   \begin{align} \nonumber |\!|\!|(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}})|\!|\!|_{1} + \| \phi^{h} - \phi \|_{ 1/2, \varGamma} \leq\;& C \frac{|s|^{2}}{\sigma \underline{\sigma}^{4}} \Big( \| s \phi \|_{1/2, \varGamma} + \| \lambda \|_{-1/2,\; \varGamma}\\ &\qquad\quad\ + \| s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}} \Big), \end{align} (5.16a)  \begin{align} \nonumber \|\lambda^{h} - \lambda \|_{-1/2, \varGamma} \leq\; & C \frac{|s|^{5/2}}{\sigma\underline{\sigma}^{7/2}} \Big(\| s \phi \|_{1/2, \varGamma} + \| \lambda \|_{-1/2,\; \varGamma} \\ &\qquad\qquad + \| s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}} \Big). \end{align} (5.16b) Regarding the proof of this result, we would like to point out only that the estimate (5.16a) follows from a combined application of (5.13) in Theorem 5.2 and (4.23) by making use of the jump condition $$\phi ^{h} -\phi =[\![\gamma{e^{h}_{v}}]\!]$$. On the other hand, in order to establish the estimate (5.16b), one has to recall that $$\lambda -\lambda ^{h} =[\![\partial _{\nu } {e^{h}_{v}}]\!]$$ and therefore an application of (4.24) combined with (5.13) yields the desired inequality. Corollary 5.3 has the awkward aspect of being an error estimate where part of the right-hand side (the terms $$\|s\phi \|_{1/2,\varGamma }$$ and $$\|\lambda \|_{-1/2,\varGamma }$$) does not converge to zero. We now clarify why this is not so. Consider the best approximation operators   $$ \varPi_{X_{h}} : H^{-1/2} (\varGamma) \mapsto X_{h} \quad \textrm{and} \quad \varPi_{Y_{h}} : H^{1/2} (\varGamma)\mapsto Y_{h}. $$If we create data for the problem so that the exact solution is $$\big (\mathbf 0,0,\varPi _{Y_{h}}\phi ,\varPi _{X_{h}}\lambda \big )$$, then the associated numerical solution will be the exact solution and there will be no error in the method. Therefore, by linearity, we can use $$\big (\mathbf u,\theta ,\phi -\varPi _{Y_{h}}\phi ,\lambda -\varPi _{X_{h}}\lambda \big )$$ as the exact solution in Corollary 5.3 and the numerical solution will be $$\big (\mathbf u^{h},\theta ^{h},\phi ^{h}-\varPi _{Y_{h}}\phi ,\lambda ^{h}-\varPi _{X_{h}}\lambda \big )$$. Consequently, we can substitute $$\|s\phi \|_{1/2,\varGamma }+\|\lambda \|_{-1/2,\varGamma }$$ by $$\| s(\phi -\varPi _{Y_{h}}\phi )\|_{1/2,\varGamma }+\|\lambda -\varPi _{X_{h}}\lambda \|_{-1/2,\varGamma }$$ in the right-hand side of (5.16a)–(5.16b). If we now apply Proposition 4.3 to Corollary 5.3, we may obtain the following estimates in the time domain. Corollary 5.4 If the exact solution quadruple satisfies   $$ \big(\mathbf U, \varTheta, \mathcal{L}^{-1} \{\phi \}, \mathcal{L}^{-1} \{\lambda \}\big ) \in W_{+}^{4}(\mathbf H^{1}(\varOmega_{-})) \times W_{+}^{4} (H^{1}(\varOmega_{-}) )\times W_{+}^{5} ( H^{1/2} (\varGamma) ) \times W_{+}^{4} ( H^{-1/2} (\varGamma) ) $$then   $$ \mathcal{L}^{-1}\big \{\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)\big \} \in \mathcal{C}\big(\mathbb{R}; \mathbf H^{1}(\varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^{1}\big(\mathbb{R}^{3} \setminus \varGamma\big)\big) $$is causal and for $$t \geq 0,$$  \begin{align*} |\!|\!|\mathcal{L}^{-1}\big\{\big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)\big\}|\!|\!|_{1} + \| \mathcal{L}^{-1} \big\{ \phi^{h} - \phi\big\} \|_{ 1/2, \varGamma} \leq \; & C t^{2}\max \big\{1, t^{4} \big\} ~g_{h}(t), \\ \|\mathcal{L}^{-1}\big\{\lambda^{h} - \lambda\big\} \|_{-1/2, \varGamma} \leq\; & C{t^{3/2}} \max \big\{1, t^{7/2} \big\}~g_{h}(t), \end{align*}where   \begin{align*} g_{h}(t) :=\; & {\int_{0}^{t}} \!\big ( \| \mathcal{P}_{4} \big( \mathcal{L}^{-1} \big\{ \dot{\phi}- \varPi_{Y_{h}} \dot{\phi}\big\}\big) (\tau) \|_{1/2, \varGamma} + \| \mathcal{P}_{4} \big( \mathcal{L}^{-1} \big\{ \lambda -\varPi_{X_{h}} \lambda \big\}\big) (\tau) \|_{-1/2, \varGamma} \big )\, \textrm{d} \tau\\ &+ {\int_{0}^{t}} \! \big ( \|\mathcal{P}_{4}\big( \ddot{ \mathbf U} - \mathbf P_{h}\ddot{\mathbf U}\big) (\tau) \|_{1, \varOmega_{-}} + \|\mathcal{P}_{4} \big( \dot{\mathbf U} - \mathbf P_{h} \dot{\mathbf U}\big) (\tau) \|_{1, \varOmega_{-}} \big )\, \textrm{d} \tau \\ &+ {\int_{0}^{t}} \! \big(\|\mathcal{P}_{4} ( \mathbf U - \mathbf P_{h}\mathbf U ) (\tau) \|_{1, \varOmega_{-}} + \| \mathcal{P}_{4} ( \varTheta - Q_{h}\varTheta ) (\tau) \|_{1, \varOmega_{-}} \big )\, \textrm{d} \tau . \end{align*} 6. Computational aspects 6.1 Convolution Quadrature We present a very brief description of the procedure used to obtain a full discretization using multistep-based Convolution Quadrature (CQ). The process was devised by Lubich (1988a,b), and was employed originally for treating convolutional boundary integral equations (Lubich & Schneider, 1992; Lubich, 1994). In recent times it has become a very powerful tool for the discretization of time-domain problems. The present description is by no means comprehensive and is provided only for the sake of completeness; the interested reader is referred to Hassell & Sayas (2016a), where very detailed descriptions of the theory and implementation for both multistep and multistage flavors of CQ are given. Suppose that   $$ \dim \,\mathbf{V}_{h} = N_{1}, \quad \dim \,W_{h} = N_{2}, \quad \dim\, Y_{h} = M_{1} \quad \textrm{and}\quad \dim \,X_{h}= M_{2}, $$and let   $$ \big\{\boldsymbol{\mu}_{j}\big\}_{j = 1}^{N_{1}},\quad \big\{ \vartheta_{j}\big\}_{j=1}^{N_{2}},\quad \big\{ \varphi_{j} \big\}_{j=1}^{M_{1}} \quad \textrm{and }\quad \big\{ \eta_{j}\big\}_{j=1}^{M_{2}} $$be the basis functions of the spaces $$\mathbf{V}_{h},\; W_{h},\; Y_{h} $$ and $$X_{h} $$, respectively. We choose a time step $$\varDelta t> 0, $$ and let us consider the uniform grid in time $$t_{n}:= n \varDelta t$$ for $$n\geq 0.$$ We define $$\mathbf A(s) \in \mathbb{C}^{(N_{1}+N_{2}+M_{1}+M_{2})\times (N_{1}+N_{2}+M_{1}+M_{2})}$$ to be the stiffness matrix of equation , where A(s) is a matrix-valued function of $$ s \in \mathbb{C}_{+}$$ whose structure is depicted in Fig. 1. Fig. 1. View largeDownload slide The linear system associated to the discretization has the block structure of the schematic. The elastic and thermal unknowns and the acoustic unknowns are weakly coupled, reflecting the physical fact that the systems communicate only through boundary interactions between the acoustic and elastic variables. Fig. 1. View largeDownload slide The linear system associated to the discretization has the block structure of the schematic. The elastic and thermal unknowns and the acoustic unknowns are weakly coupled, reflecting the physical fact that the systems communicate only through boundary interactions between the acoustic and elastic variables. The data are sampled in time and tested to define vectors $$\mathbf f_{n} \in \mathbb{R}^{(N_{1}+N_{2}+M_{1}+M_{2})}$$:   $$\begin{aligned} f_{n, i} :=\;& (D_{1}(t_{n}), \boldsymbol{\mu}_{i} )_{\varOmega_{-}}, \quad & i =& \ 1, \ldots, N_{1},\\ f_{n, i} :=\;& (D_{2}(t_{n}), \vartheta_{i})_{\varOmega_{-}}, \quad & i =& \ N_{1}+1, \ldots, N_{1}+N_{2},\\ f_{n, i} :=\;& \langle D_{3}(t_{n}), \varphi_{i}\rangle_{\varGamma}, \quad & i =& \ N_{1}+N_{2} +1, \ldots, N_{1}+N_{2} +M_{1},\\ f_{n, i} :=\; & \langle D_{4}(t_{n}), \eta_{i}, \rangle_{\varGamma}, \quad & i =& \ N_{1}+N_{2}+M_{1}+1, \ldots, N_{1}+N_{2}+M_{1}+M_{2}, \end{aligned}$$ where $$ D_{i}(t) = \mathcal{L}\{ d_{i} \}, i =1, \ldots , 4 $$ in Theorem 4.4. The CQ discretization of (5.1) starts with the Taylor expansion   \begin{align} \mathbf{A}\left( \frac{ \gamma(z)}{\varDelta t}\right) = \sum_{n=0}^{\infty} \mathbf A_{n} (\varDelta t) z^{n}, \quad \gamma (z) = \frac{\alpha_{0} + \cdots +\alpha_{k} z^{-k}}{\beta_{0} + \cdots +\beta_{k} z^{-k}}, \end{align} (6.1)where $$\gamma (z)$$ characterizes the underlying k-multistep method, and is, therefore, usually referred to as the characteristic function of the linear multistep method. For the discretization of (5.1), we seek the sequence of vectors $$\mathbf{b}_{n} \in \mathbb{R}^{(N_{1}+N_{2}+M_{1}+M_{2})}$$ given by the recurrence   \begin{align} \mathbf A_{0} (\varDelta t) ~\mathbf b_{n} = ~\mathbf f_{n} - \sum_{m=1}^{n} \mathbf A_{m} (\varDelta t)~ \mathbf b_{n-m}, \quad n \geq 0. \end{align} (6.2)The coefficients $$A_{n}(\varDelta t)$$ can be computed by means of Cauchy’s integral formula   $$ \mathbf A_{m}(\varDelta t) = \frac{1}{m!}\frac{\textrm{d}^{(m)}}{\textrm{d}z^{(m)}}\left(\mathbf A(\gamma(z)/\varDelta t)\right)|_{z=0} = \frac{1}{2\pi i}\oint_{C} \zeta^{-m-1}\mathbf A(\gamma(\zeta)/\varDelta t)\,\textrm{d}\zeta. $$For implementation purposes, the integration contour C is taken to be a circle with radius $$R_{C}$$ dependent on the number of terms in the expansion and the specific value of the computer’s machine epsilon (Hassell & Sayas, 2016a). This choice of contour allows for fast and accurate computation of the coefficients exploiting the properties of the trapezoidal rule and the fast Fourier transform. If the solution of (6.2) assumes the form $${\mathbf{{b}}}_{n} = \big (b_{n,1}, \ldots , b_{n, (N_{1}+N_{2}+M_{1}+M_{2})}\big )$$, then the Galerkin solutions of (5.1) at $$t_{n}$$ are given by   $$\begin{aligned} \mathbf{u^{h}_{n}} = \;& \sum_{j=1}^{N_{1}} b_{n,j} \boldsymbol{\mu}_{j}, & \qquad{\theta_{n}^{h}} =\;& \sum_{j=N_{1}+1}^{N_{1}+N_{2}} b_{n,j} \vartheta_{j},\\[-8pt] \nonumber\\{\phi^{h}_{n}} =\;& \sum_{j=N_{1}+N_{2}+1}^{N_{1}+N_{2}+M_{1}} b_{n,j} \varphi_{j}, &\qquad{\lambda^{h}_{n}} =\;& \sum_{j=N_{1}+N_{2}+M_{1}+1}^{N_{1}+N_{2}+M_{1}+M_{2}} b_{n, j} \eta_{j}. \end{aligned}$$ 6.2 A combined approach for time evolution The linear system arising from the discretization and depicted in Fig. 1 can be thought of as having the block structure   $$ \left[\begin{array}{cc} {\mathbf{FEM}}(s) & s\rho_{f}(\textrm N\varGamma)_{h}^\textrm{T} \\ -s\rho_{f}(\textrm N\varGamma)_{h} &{\mathbf{BEM}}(s) \end{array}\right] \left[\begin{array}{c} \left[\begin{array}{c}\mathbf u^{h} \\ \theta^{h} \end{array}\right] \\[2.5ex] \left[\begin{array}{c} \lambda^{h} \\ \phi^{h} \end{array}\right] \end{array}\right] = \left[\begin{array}{c} \left[\begin{array}{c} \!\!\!-s\rho_{f}{\varGamma_{h}^\textrm{T}}\beta^{h} \!\!\!\\ \eta^{h} \end{array}\right] \\[2.5ex] \left[\begin{array}{c} 0 \\ \rho_{f}\alpha^{h} \end{array}\right] \end{array}\right], $$where the sparse finite element block   \begin{align*} \mathbf{FEM}(s):=\;&\; s^{2}\left[\begin{array}{cc}(\rho_{\varSigma} \mathbf u_{j},\boldsymbol v_{i})_{\varOmega_{-}} & 0 \\ 0 & 0 \end{array}\right] + s\left[\begin{array}{cc} 0 & 0 \\ -(\boldsymbol\eta\mathbf u_{j}, \nabla v_{i})_{\varOmega_{-}} & (\theta_{j},v_{i})_{\varOmega_{-}}\end{array}\right] \\ &+\left[\begin{array}{cc} (\mathbf C\boldsymbol\varepsilon(\mathbf u_{j}),\boldsymbol\varepsilon(\boldsymbol v_{i}))_{\varOmega_{-}} & -(\boldsymbol\zeta\theta_{j},\boldsymbol\varepsilon(\boldsymbol v_{i}))_{\varOmega_{-}} \\ 0 & (\boldsymbol\kappa\nabla\theta_{j},\nabla v_{i})_{\varOmega_{-}}\end{array}\right] \end{align*}contains mass and stiffness matrices as well as first-order terms related to the elastic and thermal unknowns. The boundary element block BEM(s) contains the Galerkin discretization of the operators of the acoustic Calderón calculus and the coupling trace matrix $$(\textrm N\varGamma )_{h}$$ is the discretization of the bilinear form arising from the duality pairing $$\big \langle \mathbf u^{h}\cdot \boldsymbol \nu ,\chi ^{h}\big \rangle _{\varGamma }$$. Even if a CQ approach can be applied to the entire system using the expansion (6.1) on the global matrix and solving the resulting linear system (5.13), it is common practice to decouple the computations of the finite element and boundary element unknowns via a Schur complement strategy. The decoupled boundary element unknowns are then evolved in time using CQ (via (6.1) and (5.13)), while the same underlying multistep scheme is used for the finite element unknowns. This process was first described in the study by Banjai & Sauter (2009) and is explained in detail in the study by Hassell & Sayas (2016b), where it is used in the context of purely acoustic waves. 6.3 Numerical experiments In order to test numerically the formulations of the previous sections, computational convergence studies were performed in both frequency and time domains for two-dimensional test problems. Moreover, we explore numerically the case when the Lamé parameters or the thermal diffusivity and expansion are nonconstant tensors. We emphasize that the goal of the computations presented here is mainly to provide a proof of concept of the suggested discretization and to highlight the fact that the discretization can be readily implemented with only minor additions to existing code. The previous analysis remains valid in three dimensions and the implementation of the discretization in that case can be done following completely analogous steps. The computational domain. For the convergence studies, the interior domain $$\varOmega _{-}$$ where the thermoelastic equations were imposed was the polygon depicted in Fig. 2. The domain was generated and meshed using MATLAB’s pdetool. All the mesh refinements were done using the refinement capabilities of the partial differential equation toolbox. Fig. 2. View largeDownload slide Interior geometry used in the numerical experiments for both frequency-domain and time-domain studies. The domain was generated and meshed using MATLAB’s pdetool and refined uniformly using pdetool’s refinement capabilities. Fig. 2. View largeDownload slide Interior geometry used in the numerical experiments for both frequency-domain and time-domain studies. The domain was generated and meshed using MATLAB’s pdetool and refined uniformly using pdetool’s refinement capabilities. Table 1 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates (e.c.r) in the frequency domain for k = 1. The maximum length of the panels used to discretize the boundary is denoted by h k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  View Large Table 1 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates (e.c.r) in the frequency domain for k = 1. The maximum length of the panels used to discretize the boundary is denoted by h k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  View Large Approximation errors. As a measurement of the accuracy of the approximations, the difference between the manufactured solution and the approximate finite element solution was measured in the $$L^{2}(\varOmega _{-})$$- and $$H^{1}(\varOmega _{-})$$-norms for the elastic and thermal variables $$\mathbf u^{h}$$ and $$\theta ^{h}$$. For the acoustic unknown $$v^{h}$$, the approximate solution was sampled at 25 randomly placed points in $$\varOmega _{+}$$ and the maximum absolute difference from the exact solution was taken as a measure of the error. In the time-domain experiments these measurements were done for a final time t = 1.5. Table 2 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 2. The maximum length of the panels used to discretize the boundary is denoted by h k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  Table 2 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 2. The maximum length of the panels used to discretize the boundary is denoted by h k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  Table 3 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 3. The maximum length of the panels used to discretize the boundary is denoted by h k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  Table 3 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 3. The maximum length of the panels used to discretize the boundary is denoted by h k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  Table 4 Frequency-domain p-convergence studies. The experiments were run on a fixed mesh with parameter h = 0.1 using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements with k = 1, … , 5. The relative error is shown as a function of the number of degrees of freedom (Ndof) h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  Table 4 Frequency-domain p-convergence studies. The experiments were run on a fixed mesh with parameter h = 0.1 using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements with k = 1, … , 5. The relative error is shown as a function of the number of degrees of freedom (Ndof) h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  Fig. 3. View largeDownload slide Convergence studies in the frequency domain. Top row and bottom left: Successive mesh refinements were carried out for basis functions with polynomial degrees k = 1, 2 and 3. Bottom right: Approximation errors as a function of the degrees of freedom for basis functions of increasing order over a fixed mesh with parameter h = 0.1. For the color code we refer the reader to the electronic version of the manuscript. Fig. 3. View largeDownload slide Convergence studies in the frequency domain. Top row and bottom left: Successive mesh refinements were carried out for basis functions with polynomial degrees k = 1, 2 and 3. Bottom right: Approximation errors as a function of the degrees of freedom for basis functions of increasing order over a fixed mesh with parameter h = 0.1. For the color code we refer the reader to the electronic version of the manuscript. Physical parameters. The following values of the physical parameters are functions only of space and were used equally for both series of experiments. They are chosen for validation and expository purposes only and do not correspond to any relevant physical material. For the entries of the tensors we make use of the symmetries and of Voigt’s notation (Gurtin, 1973) to shorten the subscripts. 1. Density of the elastic solid and Lamé parameters:   \begin{align} \rho_{\varSigma} = 5 + \sin{(x)}\sin{(y)},\qquad \lambda = 2, \qquad \mu =3. \end{align} (6.3) 2. Table 5 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Table 5 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Table 6 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  Table 6 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  Table 7 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  Table 7 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  Table 8 Time-domain convergence results for BDF2-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  Table 8 Time-domain convergence results for BDF2-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  Thermal expansion tensor $$\boldsymbol \zeta $$:   \begin{align} \boldsymbol\zeta_{1}\leftrightarrow\, \boldsymbol\zeta_{11} \!=\! \sin{(x)}+\cos{(y)}, \quad \boldsymbol\zeta_{2} \leftrightarrow \,\boldsymbol\zeta_{22} \!=\! -\sin{(y)}, \quad \boldsymbol\zeta_{3} \leftrightarrow\, \boldsymbol\zeta_{12} \!=\! \boldsymbol\zeta_{21} \!=\! \cos{(x)}. \end{align} (6.4) 3. Thermal diffusivity tensor $$\boldsymbol \kappa $$:   \begin{align} \boldsymbol\kappa_{1}\leftrightarrow\, \boldsymbol\kappa_{11} = 10 + x^{2}, \quad \boldsymbol\kappa_{2} \leftrightarrow \,\boldsymbol\kappa_{22} = 10+ y, \quad \boldsymbol\kappa_{3} \leftrightarrow\, \boldsymbol\kappa_{12}= \boldsymbol\kappa_{21} = 0. \end{align} (6.5) 4. The components of the tensor $$\boldsymbol \eta $$ were chosen to be   \begin{align} \boldsymbol\eta_{1}\leftrightarrow\, \boldsymbol\eta_{11} = 1, \quad \boldsymbol\eta_{2} \leftrightarrow \,\boldsymbol\eta_{22} = x + y, \quad \boldsymbol\eta_{3} \leftrightarrow\, \boldsymbol\eta_{12}= \boldsymbol\eta_{21} = 5+x+y. \end{align} (6.6) Convergence studies in the frequency domain. We first verify the results in the frequency domain. We proceed by the method of manufactured solutions using the functions  \begin{align*} \mathbf u :=&\, \big(x^{3}+xy+y^{3},\sin{(x)}\cos{(y)}\big), \qquad \theta := \sin^{2}{(\pi x)}\sin^{2}{(y)}, \\ v :=&\, \tfrac{i}{4}H_{0}^{(1)}(isr), \qquad \qquad\qquad\qquad\quad\;\;\, r = \sqrt{x^{2}+y^{2}}, \end{align*}together with the parameters defined in (6.3)–(6.6). Right-hand-side load vectors and boundary conditions were constructed accordingly. For the numerical experiments, Lagrangian $$\mathcal P_{k}$$ finite elements were used for the elastic and thermal unknowns, while Galerkin $$\mathcal P_{k}/\mathcal P_{k-1}$$ continuous/discontinuous boundary elements were used for the acoustic potential v. Convergence studies for spatial refinements with a fixed polynomial degree (h-convergence) and increasing degree of polynomial approximation with a fixed mesh size (p-convergence) were performed for s = 2.8i. The results of the mesh-refinement experiments are shown in Tables 1, 2 and 3. Table 4 contains the results for a fixed mesh with increasing polynomial degree for the basis functions. The convergence plots for all the simulations are displayed in Fig. 3. Convergence studies in the time domain. In a way analogous to the previous section, the numerical experiments were carried out using the physical parameters and coefficients given in (6.3)–(6.6) and with manufactured solutions using the functions  \begin{align*} \mathbf u :=&\, \textrm T(t) \big(x^{3}+xy+y^{3},\sin{(x)}\cos{(y)}\big), \qquad \theta := \textrm T(t) \sin^{2}{(\pi x)}\sin^{2}{(y)}, \\ v :=&\,\mathcal{L}^{-1}\left\{iH^{(1)}_{0}(i s r)\,\mathcal{L}\{\mathcal{H}(t)\sin(3t)\} \right\}, \qquad r := \sqrt{x^{2}+y^{2}}, \end{align*} Fig. 4. View largeDownload slide Time-domain convergence studies for the BDF2-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Fig. 4. View largeDownload slide Time-domain convergence studies for the BDF2-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Table 9 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  View Large Table 9 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  View Large Table 10 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  View Large Table 10 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  View Large Table 11 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  View Large Table 11 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  View Large Table 12 Time-domain convergence results for trapezoidal-rule-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  View Large Table 12 Time-domain convergence results for trapezoidal-rule-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  View Large Fig. 5. View largeDownload slide Time-domain convergence studies for the trapezoidal rule-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Fig. 5. View largeDownload slide Time-domain convergence studies for the trapezoidal rule-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. where $$\mathcal L\{\cdot \}$$ is the Laplace transform, the time factor T(t) is given by   \begin{align} \textrm T := \mathcal H(t)\big(t^{2}+2t\big), \end{align} (6.8a)and $$\mathcal H(t)$$ is the $$\mathcal C^{5}$$ approximation to Heaviside’s step function   $$ \mathcal H(t)\!:= \!t^{5}(1-5(t-1)\!+\!15(t-1)^{2}\!-\!35(t-1)^{3}\!+\!70(t-1)^{4}\!-\!126(t-1)^{5})\chi_{[0,1]}(t)\!+\!\chi_{[1,\infty)}(t). $$Two kinds of experiments were carried out using the same geometry as in the frequency domain. First, for a spatial discretization with fixed polynomial degree, successive dyadic refinements in both mesh size h and time step $$\varDelta t$$ were carried out (h-refinement). The experiment was repeated for polynomial degrees k = 1, 2 and 3 starting with a spatial mesh with parameter $$h=1 \times 10^{-1}$$ and time step $$\varDelta t = 3.75 \times 10^{-2}$$. The second experiment corresponds to p-refinements in space and consisted in using a fixed spatial mesh, starting with a polynomial discretization of degree k = 1 in space and a time step $$\varDelta t = 3.75 \times 10^{-2}$$. With every successive dyadic refinement of $$\varDelta t$$, the degree of the polynomial interpolant was increased by 1. The initial mesh size of $$h=5.016 \times 10^{-2}$$ corresponds to the second level of refinement used for the h-refinement experiments. These space-time refinement strategies highlight the global order of convergence of the method, which is expected to be asymptotically limited by the order of the multistep scheme used for time discretization. Both strategies were tried for BDF2 and trapezoidal-rule-based time discretizations. The results for the studies using BDF2 are shown in Tables 5, 6, 7 (h-refinement) and 8 (p-refinement). These results are summarized in the convergence plot of Fig. 4. Similarly, the results for the experiments using trapezoidal-rule time stepping are shown in Tables 9, 10, 11 (h-refinement) and 12 (p-refinement), all condensed on the convergence plots shown in Fig. 5. Table 13 Number of finite element degrees of freedom required to represent a single scalar function defined on the mesh depicted in Fig. 2. The number increases depending on the chosen refinement strategy, with the slowest growth rate being the one associated to p-refinements Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Table 13 Number of finite element degrees of freedom required to represent a single scalar function defined on the mesh depicted in Fig. 2. The number increases depending on the chosen refinement strategy, with the slowest growth rate being the one associated to p-refinements Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Fig. 6. View largeDownload slide Snapshots at times t = 0.6, 1.3, 2 of the interaction between an acoustic wave and a thermoelastic obstacle. The acoustic field is depicted in the top row, the norm of the elastic displacement in the middle row (black denotes no displacement) and the temperature variations in the bottom row (black represents zero variation, shades of red and blue represent positive and negative variations, respectively). The reader is referred to the on-line version of the paper for the color code. Fig. 6. View largeDownload slide Snapshots at times t = 0.6, 1.3, 2 of the interaction between an acoustic wave and a thermoelastic obstacle. The acoustic field is depicted in the top row, the norm of the elastic displacement in the middle row (black denotes no displacement) and the temperature variations in the bottom row (black represents zero variation, shades of red and blue represent positive and negative variations, respectively). The reader is referred to the on-line version of the paper for the color code. Depending on the refinement strategy, the number of degrees of freedom (DOF) required to approximate the system increases quickly, especially for h-refinements with a higher-order polynomial basis. Table 13 shows the number of unknowns associated to a single scalar FEM function represented in the grid shown in Fig. 2. The increase in computational requirements imposed by h-refinement makes some asymptotic properties of the scheme difficult to observe following this strategy. In particular, the smoothing properties of the parabolic part of the system introduce superconvergent behavior on the thermal unknowns during the pre-asymptotic regime. As can be seen in the p-refinement experiments (cf. Figs 4 and 5, bottom right) the convergence stabilizes to the predicted rate for relatively small time steps, after 5 refinement levels. The number of spatial degrees of freedom required to achieve such a discretization level by h-refinements causes the true convergence rate to be observable using only a p-refinement strategy. One example. We conclude with a simple illustrative example in two dimensions showing the interaction between the plane wave   $$ v^{\textrm{inc}}= 3\chi_{[0,0.3]}(88\tau)\sin{(88\tau)},\quad \tau:= t-\mathbf r\cdot\mathbf d,\quad \mathbf r:=(x,y),\quad \mathbf d := (1,5)/\sqrt{26}, $$and a pentagonal scatterer with mass density given by   $$ \rho_{\varSigma} = 15 + 40e^{-49\, r^{2}}, \qquad r := \sqrt{x^{2}+y^{2}}. $$ The values of the elastic parameters, thermal diffusivity $$\boldsymbol \kappa $$, thermoelastic expansion tensors $$\boldsymbol \zeta $$ and $$\boldsymbol \eta $$ were the same as those used for the convergence experiments in the previous paragraphs and given in equations (6.3)–(6.6). The simulation used $$\mathcal P_{2}$$ Lagrangian finite elements on a grid with mesh parameter $$h=7 \times 10^{-3}$$ and 36096 elements. The inherited boundary element grid had 496 panels and a grid parameter of $$h=9.1 \times 10^{-3}$$, and $$\mathcal P_{2}/\mathcal P_{1}$$ continuous/discontinuous Galerkin boundary elements were used. Trapezoidal-rule-based discretization was applied in time with a time step $$\varDelta t=1\times 10^{-2}$$. Some snapshots of the simulation are shown in Fig. 6. Acknowledgements The authors would like to thank the referees for their detailed comments and suggestions, which greatly improved the quality of this communication. References Bamberger, A. & Ha-Duong, T. ( 1986) Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique (I). Math. Meth. Appl. Sci. , 8, 405– 435. Google Scholar CrossRef Search ADS   Banjai, L., Lubich, C. & Sayas F. J. ( 2015) Stable numerical coupling of exterior and interior problems for the wave equation. Numer. Math.,  129, 611– 646. Google Scholar CrossRef Search ADS   Banjai L. & Sauter, S. ( 2008/09) Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. , 47, 227– 249. Google Scholar CrossRef Search ADS   Beltrami, E. J. & Wohlers, M. R. ( 1966) Distributions and the Boundary Values of Analytic Functions . New York-London: Academic Press, p. xiv+116. Çakoni, F. ( 2000) Boundary integral method for thermoelastic screen scattering problem in $$\mathbb{R}^3$$. Math. Methods Appl. Sci. , 23, 441– 466. Google Scholar CrossRef Search ADS   Çakoni F. & Dassios, G. ( 1998) The coated thermoelastic body within a low-frequency elastodynamic field. Int. J. Eng. Sci. , 36, 1815– 1838. Google Scholar CrossRef Search ADS   Carlson, D. ( 1972) Linear Thermoelasticity . Encyclopedia of Physics (C. Truesdell ed.), Vol. VIa/2. New York: Springer. Dassios, G. & Kostopoulos, V. ( 1994) Scattering of elastic waves by a small thermoelastic body. Int. J. Eng. Sci. , 32, 1593– 1603. Google Scholar CrossRef Search ADS   Dautray, R. & Lions, J.-L. ( 1992) Mathematical Analysis and Numerical Methods for Science and Technology. Evolution Problems I , vol. 5. Berlin: Springer. With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon (Translated from the French by Alan Craig). Duhamel, J.-M.-C. ( 1837) Second mémoire sur les phénomènes thermo-mécaniques. J. de l’École Polytechnique , 15, 1– 57. Fichera, G. ( 1973) Existence theorems in elasticity . Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells , vol. 4. Berlin, Heidelberg: Springer, pp. 347– 389. Gurtin, M. E. ( 1973) The linear theory of elasticity. Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells (C. Truesdell ed.). Berlin, Heidelberg: Springer, pp. 1– 295. Hahn, D. W. & Ozisik, M. N. ( 2012) Heat Conduction . Hoboken, New Jersey: John Wiley. Google Scholar CrossRef Search ADS   Hassell, M.E. & Sayas F. -J. ( 2016a) Convolution quadrature for wave simulations. Numerical Simulation in Physics and Engineering. Volume 9 of SEMA SIMAI Springer Series.  Cham: Springer. pp. 71– 159. Hassell, M.E. & Sayas F.-J. ( 2016b) A fully discrete BEM–FEM scheme for transient acoustic waves. Comput. Methods Appl. Mech. Engrg. , 309, 106– 130. Google Scholar CrossRef Search ADS   Hsiao, G. C., Sánchez-Vizuet, T. & Sayas, F. -J. ( 2016) Boundary and coupled boundary–finite element methods for transient wave–structure interaction. IMA J. Numer. Anal. , 37, 237– 265. Google Scholar CrossRef Search ADS   Hsiao, G. C., Sayas, F. J. & Weinacht, R. J. ( 2013) A time-dependent fluid–structure interaction. Math. Meth. Appl. Sci. , DOI: https://doi.org/10.1002/sim.0000. Hsiao, G. C. & Wendland, W. L. ( 2008) Boundary Integral Equations . 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Google Scholar CrossRef Search ADS   Landau, L. D. & Lifshitz, E. M. ( 1986) Course of Theoretical Physics , vol. 7, 3rd edn. Theory of elasticity, Translated from the Russian by J.B. Sykes and W.H. Reid. Oxford: Pergamon Press, pp. viii+187. Lin, W. H. & Raptis, A. C. ( 1983) Thermoviscous effects on acoustic scattering by thermoelastic solid cylinders and spheres. J. Acoust. Soc. Am ., 74, 1542– 1554. Google Scholar CrossRef Search ADS   Lopat’ev, A. A. ( 1979) Effect of thermoelastic scattering in a liquid and solid body on the reflection of harmonic waves from a plane boundary of separation. Sov. Appl. Mech. , 15, 79– 82. Google Scholar CrossRef Search ADS   Lubich, Ch. ( 1988a) Convolution quadrature and discretized operational calculus. I. Numer. Math. , 52, 129– 145. Google Scholar CrossRef Search ADS   Lubich, Ch. ( 1988b) Convolution quadrature and discretized operational calculus. II. Numer. Math. , 52, 413– 425. Google Scholar CrossRef Search ADS   Lubich, Ch. 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Sayas, F.-J ( 2016) Retarded Potentials and Time Domain Boundary Integral Equations: A Road Map . Springer Series in Computational Mathematics , vol. 50. Springer, Cham: Springer, pp. xv+242. Google Scholar CrossRef Search ADS   Wagner, P. ( 1994) P. Fundamental matrix of the system of dynamic linear thermoelasticity. J. Therm. Stresses , 17, 592– 565. Google Scholar CrossRef Search ADS   © The Author(s) 2018. 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/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

A time-dependent wave-thermoelastic solid interaction

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Abstract

Abstract This paper presents a combined field and boundary integral equation method for solving the time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid–structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analysed with Lubich’s approach for time-dependent boundary integral equations. Existence and uniqueness results are established in terms of time-domain data with the aid of Laplace domain techniques. Galerkin semidiscretization approximations are derived and error estimates are obtained. A full discretization based on the convolution quadrature method is also outlined. Some numerical experiments in two dimensions are also included in order to demonstrate the accuracy and efficiency of the procedure. 1. Introduction The mathematical study of the thermodynamic response of a linearly elastic solid to mechanical strain dates back at least to Duhamel’s (1837) pioneering work on thermoelastic materials where he proposed the constitutive relation linking the temperature variations and elastic strains with the thermoelastic stress now known as the Duhamel–Neumann law (Carlson, 1972; Maugin, 2014). Kupradze’s (1979) encyclopedic works can be considered the standard reference for a modern mathematical treatment of the purely thermoelastic problem. The dynamic problem is dealt with in more recent works like Ortner & Wagner (1992) and Wagner (1994) where the matrix of fundamental solutions for the dynamic equations is revisited, while Jakubowska (1982, 1984) provide generalized Kirchhoff-type formulas for thermoelastic solids. In the case of the scattering of thermoelastic waves, major theoretical contributions have been made by Çakoni & Dassios (1998). The unique solvability of a boundary integral formulation is established in the study by Çakoni (2000) and the interaction of elastic and thermoelastic waves is explored for homogeneous materials in the study by Dassios & Kostopoulos (1994). The study of time-harmonic interaction between a scalar field and a thermoelastic solid has been the subject of works like Lopat’ev (1979) where the interface is taken to be a plane, or Lin & Raptis (1983) and Jentsch & Natroshvili (1997) where time-harmonic scattering by bounded obstacles is considered. In this paper we present a combined field and boundary integral method for a time-dependent fluid–thermoelastic solid interaction problem. The approach here is a generalization of the method employed by the authors for treating time-dependent fluid–structure interaction problems in Hsiao et al. (2016) and Hsiao et al. (2013). The present communication is an improvement over those previous efforts in the sense that it considers a more general constitutive law that accounts for the coupling between elastic and thermal effects. To our knowledge, other than the preliminary results in Sánchez-Vizuet (2016), no attempt has been made to investigate with rigorous justifications the time-dependent acoustic scattering by a thermoelastic obstacle. The setting is introduced in Section 2, along with the physical assumptions and the constitutive relation under consideration leading to the time-domain system of governing equations. The problem is then recast in Section 3, where the Laplace domain system is transformed into an equivalent integro-differential nonlocal problem that will be formulated variationally for discretization later on. The question of existence and uniqueness of the solutions to the nonlocal problem is dealt with in Section 4. The error analysis of the proposed discretization is addressed in Section 5, where semidiscrete error estimates for spatial discretization are provided. The final Section 6 discusses the computational considerations related to the numerical solution of the discrete problem. A full discretization using convolution quadrature (CQ) in time is outlined and the coupling of boundary and finite elements for the spatial discretization is discussed. Convergence experiments in two dimensions are performed for test problems in both frequency and time domains as a demonstration of the applicability of the formulation, which remains valid also in three dimensions. For time discretization, both second-order backward differentiation formula (BDF2) and trapezoidal rule CQ are used, providing evidence that the approximation is stable and of second order globally. One simple illustrative experiment in the time domain using the proposed formulation is included to demonstrate the applicability of the procedure. In closing, we remark that for a homogeneous thermoelastic solid medium, a pure boundary integral equation formulation may be adapted as in the fluid–structure interaction problem (Hsiao et al., 2016). We will pursue these investigations in a separate communication. 2. Formulation of the problem Consider a thermoelastic solid with constant density $$\rho _{\varSigma }$$ in an undeformed reference configuration and at thermal equilibrium at temperature $$\varTheta _{0}$$. Under the action of external forces the body will be subject to internal stresses that will induce local variations of temperature. Reciprocally, if a heat source induces a change in temperature, the body will react by dilating or contracting and this will create internal stresses and deformations. We will denote by U the elastic deformation with respect to the reference configuration and by $$\varTheta $$ the variation of temperature with respect to the equilibrium temperature. In the classical linear theory (Kupradze, 1979; Landau et al., 1986), the coupling between the mechanical strain and the thermal gradient is modeled by the Duhamel–Neumann law. This constitutive relation defines the thermoelastic stress$$\boldsymbol \sigma (\mathbf U,\varTheta )$$ and the thermoelastic heat flux$$\mathbf F(\mathbf U,\varTheta )$$ (also known as free energy) as functions of the elastic displacement and the variation in temperature through the equations  \begin{align*} \boldsymbol\sigma :=\,& \mathbf C\varepsilon(\mathbf U) - \zeta\varTheta\mathbf I\,,\\ \mathbf F :=\,& -\eta\,\frac{\partial \mathbf U}{\partial t} + \kappa \nabla\varTheta. \end{align*} In the previous expressions,   $$ \varepsilon (\mathbf{U}) := \tfrac{1}{2}\left( \nabla\mathbf{U} + \left(\nabla\mathbf{U}\right)^{\textrm T}\right) $$is the elastic strain tensor, I is the 3 × 3 identity matrix, $$\kappa $$ is the thermal diffusivity coefficient, which from physical principles (Hahn & Ozisik, 2012) is required to be positive, $$\zeta $$ is the product of the volumetric thermal expansion coefficient and the bulk modulus of the material and $$\eta $$ is given by the relation   $$ \eta = \varTheta_{0}\zeta/c_{\textrm{vol}}. $$Here the volumetric heat capacity $$c_{\textrm{vol}}$$ is the ratio between the thermal diffusivity and the thermal conductivity, and it can also be expressed as the product of the mass density and the specific heat capacity. For the case of a homogeneous isotropic material that we are considering, the elastic stiffness tensorC is given by   $$ \mathbf C_{ijkl}:= \lambda\delta_{ij}\delta_{kl} +\mu\left(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}\right), $$where the constants $$\lambda $$ and $$\mu $$ are Lamé’s second parameter and the shear modulus, respectively, and $$\delta _{ij}$$ is Kronecker’s delta. We are concerned with a time-dependent direct scattering problem in fluid–thermoelastic solid interaction, which can be simply described as follows: an acoustic wave propagates in a fluid domain of infinite extent in which a bounded thermoelastic body is immersed. Throughout the paper, we let $$\varOmega _{-}$$ be the bounded domain in $$\mathbb{R}^{3}$$ occupied by the thermoelastic body with a Lipschitz boundary $$\varGamma $$ and we let $$\varOmega _{+} := \mathbb{R}^{3} \setminus \overline{\varOmega }_{-}$$ be its exterior, occupied by a compressible fluid. The problem is then to determine the scattered velocity potential V in the fluid domain, the deformation of the solid U and the variation of the temperature $$\varTheta $$ in the obstacle. It is assumed that $$|\varTheta /\varTheta _{0}|\ll1$$. The governing equations of the displacement field U and temperature field $$\varTheta $$ are the thermo-elastodynamic equations:  \begin{align} \rho_{\varSigma} \frac{\partial^{2}\mathbf{U}}{\partial t^{2}} - \varDelta^{*} \mathbf{U} + \zeta\, \nabla\varTheta = \mathbf{0} \quad \textrm{in}\; \varOmega_{-} \times (0,T), \end{align} (2.1)  \begin{align} \frac{1}{\kappa}\frac{\partial \varTheta}{\partial t} - \varDelta \varTheta + \eta\; \frac{\partial}{\partial t}(\nabla\cdot\mathbf{U}) = 0 \quad \textrm{in}\; \varOmega_{-} \times (0,T), \end{align} (2.2)where T is a given positive final time, and as usual the symbol $$\varDelta ^{\ast }$$ is the Lamé operator defined by   $$ \varDelta^{\ast} \mathbf{U} := \mu \varDelta \mathbf{U} + (\lambda + \mu) \nabla(\nabla\cdot \mathbf U). $$We remark that if the thermal effect is neglected ($$\zeta =0$$), Duhamel–Neumann’s law reduces to the usual expression for Hooke’s law of the classical theory for an arbitrary isotropic medium (see, e.g., Carlson, 1972 and Kupradze, 1979). In the thermoelastic medium, the given physical constants $$ \rho _{\varSigma }, \lambda , \mu , \zeta , \eta , \kappa $$, are assumed to satisfy the inequalities   $$ \rho_{\varSigma}> 0, \;\;\mu > 0, \;\;3 \lambda + 2 \mu > 0, \;\;\frac{\zeta}{\eta} > 0, \;\;\kappa > 0. $$In the fluid domain $$\varOmega _{+}$$, we consider a barotropic and irrotational flow of an inviscid and compressible fluid with density $$\rho _{f}$$ as in the study by Hsiao et al. (2013). The formulation can be presented in terms of a scalar potential V = V (x, t) such that the scattered velocity field V and the pressure P are given by   $$ \mathbf{V} = - \nabla\; V \quad\textrm{and}\quad P={\rho_{f}} \frac{\partial V}{\partial t}. $$We thus arrive at the wave equation   \begin{align} \frac{1}{c^{2}} \frac{\partial^{2} V}{\partial t^{2}} - \varDelta V = 0 \quad\textrm{in}\quad \varOmega_{+} \times (0, T), \end{align} (2.3)where c is the sound speed. On the interface $$\varGamma $$ between the solid and the fluid we have the transmission conditions   \begin{align} \boldsymbol\sigma(\mathbf U, \varTheta)^{-}\mathbf{n} =\, -{\rho_{f}}\;\left( \frac{\partial V}{\partial t} + \frac{\partial V^{\textrm{inc}}}{\partial t}\right)^{+}\mathbf{n} \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4a)  \begin{align} \frac{ \partial\mathbf{U}^{-}}{\partial t} \cdot\mathbf{n} = \, - \left(\frac{\partial V}{\partial n} + \frac{\partial V^{\textrm{inc}}}{\partial n}\right)^{+} \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4b)  \begin{align} \frac{\partial \varTheta}{\partial n}^{-} = \, 0 \quad \textrm{on}\;\varGamma \times (0, T), \end{align} (2.4c)where n is the exterior unit normal to $$\varOmega _{-}$$, and $$V^{\textrm{inc}} $$ denotes the given incident field, which is assumed to be supported away from $$\varGamma $$ at t = 0. Here and in the sequel, we adopt the notation that $$q^{\mp }$$ denotes the limit of the function q on $$\varGamma $$ from $$\varOmega _{\mp }$$, respectively. Regarding the transmission conditions we remark that from the physical point of view, equation (2.4a) enforces the equilibrium of pressure at the solid–fluid interface, condition (2.4b) expresses the continuity of the normal component of the velocity field, and (2.4c) refers to a thermally insulated body. We assume the causal initial conditions   \begin{align} \mathbf{U}(x,t) = \frac{\partial\mathbf{U}(x,t)}{\partial t} =\mathbf{0}, \quad \varTheta(x, t) = 0 \quad \textrm{for}\quad x \in \varOmega_{-},\;\; t \leq 0, \end{align} (2.5a)  \begin{align} V(x,t) = \frac{\partial V}{\partial t} (x,t) = 0\;\; \textrm{for} \quad x\in \varOmega_{+},\;\; t \leq 0. \end{align} (2.5b)We will study the time-dependent scattering problem consisting of the partial differential equations (2.1)--(2.3) together with the transmission conditions (2.4a)–(2.4c) and the homogeneous initial conditions (2.5a)–(2.5b). 3. Reduction to a nonlocal problem on a bounded domain In order to apply Lubich’s approach as in the case of fluid–structure interaction (Hsiao et al., 2013; Hsiao et al., 2016), we first need to transform the initial–boundary transmission problem (2.1)–(2.4c) in the Laplace domain. To do that we first reduce the corresponding problem to a nonlocal boundary value problem. We begin with the Laplace transform for a restricted class of distributions. Let X be a Banach space and $$\mathcal S(\mathbb R)$$ denote the Schwartz class of functions. We say that $$F: \mathcal S(\mathbb R)\rightarrow X$$ is a causal tempered distribution with values in X if it is a continuous linear map such that   $$ F(\varphi) = 0 \quad \forall\, \varphi\in\mathcal S (\mathbb R) \textrm{ such that } \textrm{supp}\;\varphi \subset (-\infty,0). $$For such a distribution and for   $$ s\in\mathbb C_{+} :=\big\{ s \in \mathbb C : \textrm{Re}\;s> 0\big\}, $$the Laplace transform of F can be defined in a natural way by   $$ f(s)= \mathcal{L}\big\{F\big\}(s) := \int_{0}^{\infty} e^{-st} F(t)\, \textrm{d}t, $$where the integral must be understood in the sense of Bochner (Sayas, 2016) when F is integrable or as a duality product in the general case. We remark that the Laplace transform can be defined for a much broader class of distributions (Beltrami & Wohlers, 1966; Dautray & Lions, 1992), but this restricted class suffices for the current application. Then let   $$ \mathbf{u}:= \mathbf{u}(x,s)= \mathcal{L}\big\{\mathbf{U}(x,t)\big\}, \;\; \theta:=\theta(x,s) = \mathcal{L}\big\{{\varTheta(x,t)} \big\},\;\; v:=v(x,s) = \mathcal{L}\big\{V(x,t)\big\}. $$The initial–boundary transmission problem consisting of (2.1)–(2.4c) in the Laplace transformed domain becomes the following transmission boundary value problem:   \begin{align} {\hskip-76pt} - \varDelta^{*} \mathbf{u} + \rho_{\varSigma} s^{2} \mathbf{u} + \zeta \nabla\theta = \, \mathbf{0} \quad \textrm{in}\quad \varOmega_{-}, \end{align} (3.1a)  \begin{align} {\hskip-77.5pt} - \varDelta \theta + \frac{s}{\kappa} ~\theta + s~ \eta ~\nabla\cdot \mathbf{u} =\, 0 \quad \textrm{in} \quad \varOmega_{-}, \end{align} (3.1b)  \begin{align}{\hskip-31.5pt} -\varDelta v + \frac{s^{2}}{c^{2}} \;v = \, 0 \quad \textrm{in}\quad \varOmega_{+}, \end{align} (3.1c)  \begin{align}{\hskip-31pt} \boldsymbol{\sigma} (\mathbf{u}, \theta )^{-} \mathbf{n} + \rho_{f}\; s v^{+} \mathbf{n} =\, - \rho_{f}s v^{\textrm{inc}}\mathbf{n} \quad \textrm{on}\quad \varGamma, \end{align} (3.1d)  \begin{align} {\hskip-19pt} s \; \mathbf{u}^{-} \cdot \mathbf{n} + \frac{ \partial v}{\partial n}^{+} =\, -\frac{\partial v^{\textrm{inc}}}{\partial n} \quad \textrm{on}\quad \varGamma, \end{align} (3.1e)  \begin{align} \frac{\partial \theta}{\partial n}^{-} = \, 0 \quad \textrm{on} \quad \varGamma. \end{align} (3.1f)We remark that (3.1) is an exterior scattering problem for which normally a radiation condition is needed in order to guarantee the uniqueness of the solution. However, in the present case no additional radiation condition is required and global $$H^{1}$$ behavior at infinity suffices; uniqueness then follows from a standard energy argument and the fact that Re s > 0. To derive a proper nonlocal boundary problem, as usual, we begin via Green’s third identity with the representation of the solutions of (3.1c) in the form:   \begin{align} v = \mathcal{D}(s){\phi} - \mathcal{S}(s){\lambda} \quad \textrm{in} \quad \varOmega_{+}, \end{align} (3.2)where $${\phi }:= v^{+}$$ and $${\lambda }:= \partial v^{+} /\partial n$$ are the Cauchy data for v in (3.1c) and $$\mathcal{S}(s)$$ and $$\mathcal{D}(s)$$ are the simple-layer and double-layer potentials, respectively, defined by   \begin{align} \mathcal{S}(s){\lambda} (x) :=\, \int_{\varGamma} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y},\qquad x \in \mathbb R^{3}\setminus\varGamma, \end{align} (3.3)  \begin{align} \mathcal{D}(s){\phi} (x) :=\, \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y},\quad x \in \mathbb R^{3}\setminus\varGamma, \end{align} (3.4)with respect to the complex frequency $$s\in \mathbb C_{x+}$$, where   $$ E_{s/c}(x,y) = \frac{e^{- s/c\; |x-y|}}{4 \pi |x-y|} $$is the fundamental solution of the operator in (3.1c). By standard arguments in potential theory, we have the relations for the Cauchy data $${\lambda }$$ and $${\phi } $$,   \begin{align} \left(\begin{array}{cccccc} {\phi} \\[3mm] {\lambda}\\ \end{array}\right) = \left ( \begin{matrix} \frac{1}{2}I + K(s) & -V(s) \\[3mm] -W(s) & \frac{1}{2}I - K^{\prime}(s) \\ \end{matrix} \right )\left(\begin{array}{cccccc} {\phi} \\[3mm] {\lambda}\\ \end{array}\right) \quad \textrm{on} \quad \varGamma. \end{align} (3.5)Here $$V, K, K^{\prime } $$ and W are the four basic boundary integral operators familiar from potential theory (Hsiao & Wendland, 2008) such that   \begin{align*} V(s) \lambda (x) :=\,& \int_{\varGamma} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\quad\ \ x \in \varGamma,\\ K(s){\phi} (x) :=\,& \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\! x \in \varGamma,\\ K^{\prime}(s)\lambda (x) :=\,& \int_{\varGamma} \frac{\partial}{\partial n_{x}} E_{s/c}(x,y){\lambda}(y)\, \textrm{d}\varGamma_{y}, \qquad\qquad\quad\! x \in \varGamma,\\ W(s){\phi} (x) :=\,&- \frac{\partial}{\partial n_{x}} \int_{\varGamma} \frac{\partial}{\partial n_{y}} E_{s/c}(x,y){\phi}(y)\, \textrm{d}\varGamma_{y}, \qquad x \in \varGamma. \end{align*}The operator $$K^{\prime }(s)$$ is the real transpose of K(s), i.e., it is the conjugate of the Hilbert space adjoint. Note that all our Hilbert spaces are complexified versions of real Hilbert spaces and real adjoints/transposes can be defined. By using the transmission condition (3.1e), we obtain from the second boundary integral equation in (3.5),   \begin{align} -s \mathbf{u}^{-} \cdot \mathbf{n} + W(s){\phi} - \left( \tfrac{1}{2} I - K^{\prime}(s)\right) {\lambda} = \frac{\partial v^{\textrm{inc} }}{\partial n} \;\quad \textrm{on}\quad \varGamma, \end{align} (3.6)while the first boundary integral equation in (3.5) is simply  \begin{align} \left( \tfrac{1}{2}I - K(s)\right){\phi} + V(s){\lambda} = 0 \quad \textrm{on} \quad \varGamma. \end{align} (3.7)With the Cauchy data $$\phi $$ and $$\lambda $$ as new unknowns, the partial differential equation (3.1c) in $$\varOmega _{+}$$ may be eliminated. This leads to a nonlocal boundary value problem in $$\varOmega _{-}$$ for the unknowns $$(\mathbf u, \theta , \phi , \lambda )$$ satisfying the partial differential equations (3.1a), (3.1b), and the boundary integral equations (3.6), (3.7) together with the conditions (3.1d) and (3.1f) on $$\varGamma $$. Here and in the sequel let $$\gamma ^{\mp } $$ and $$\partial _{n}^{\mp }$$ denote trace operators of the functions and their normal derivatives from inside and outside $$\varGamma $$, respectively. We will use the symbol $$(\cdot ,\cdot )_{\mathcal{O}}$$ interchangeably to denote the scalar, vector or matrix $$L^{2}$$ inner products of functions defined on the open set $$\mathcal O$$, while the angled brackets $$\langle \cdot ,\cdot \rangle _{\varGamma }$$ will be reserved for pairings between elements of the trace space and its dual. All the forms will be kept bilinear and conjugation will be done explicitly when needed. Finally, the space $$\mathbf H^{1}(\mathcal O)$$ should be understood as the Cartesian product of copies of the standard scalar Sobolev space $$H^{1}(\mathcal O)$$ endowed with the natural product norm. Let us first consider the unknowns $$(\mathbf u,\theta ) \in{\mathbf H}^{1}(\varOmega _{-})\times H^{1}(\varOmega _{-})$$. Multiplying (3.1a) by the testing function v and integrating by parts, we obtain the weak formulation of (3.1a):   \begin{align} a({\mathbf u},{\mathbf v}; s) - \langle{\boldsymbol \sigma}(\mathbf u,\theta)\mathbf n, \gamma^{-}{\mathbf v}\rangle_{\varGamma} - \zeta(\theta, \; \nabla\cdot\mathbf v)_{\varOmega_{-}} = 0, \end{align} (3.8)where   \begin{align} a(\mathbf u, \mathbf v; s): = \left( \mathbf C\boldsymbol \varepsilon (\mathbf u), \boldsymbol \varepsilon ( \mathbf v)\right)_{\varOmega_{-}} + s^{2}\rho_{\varSigma}( \mathbf u, \mathbf v)_{\varOmega_{-}}. \end{align} (3.9)In terms of the transmission condition (3.1d), we obtain from (3.8),   \begin{align} a({\mathbf u},{\mathbf v}; s) - \zeta (\theta, \; \nabla\cdot\mathbf v)_{\varOmega_{-}} + \rho_{f} s \langle \phi~{\mathbf n}, \gamma^{-}{\mathbf v}\rangle_{\varGamma} = - s\rho_{f} \langle v^{\textrm{inc}} \mathbf n, \gamma^{-} \mathbf v \rangle_{\varGamma}. \end{align} (3.10)Similarly, multiplying (3.1b) by the test function $$\vartheta $$, integrating by parts and making use of the condition (3.1f), we have   \begin{align} b( \theta, \vartheta; s) + s\; \eta(\nabla\cdot\mathbf u, \vartheta)_{\varOmega_{-}}= 0 \end{align} (3.11)with   \begin{align} b(\theta, \vartheta;s) := (\nabla \theta, \nabla \vartheta)_{\varOmega_{-}} + \frac{s}{\kappa} ( \theta, \vartheta)_{\varOmega_{-}}. \end{align} (3.12) Now let   $$\begin{aligned} \mathbf A_{s}:\;& \mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (\mathbf H^{1}(\varOmega_{-}))^{\prime}\!, \\ &\;\mathbf u \;&\; \longmapsto \;&\; a(\mathbf u, \,\cdot\,;s), \\ B_{s} :\;& H^{1}(\varOmega_{-})\;& \;\longrightarrow\;&\; (H^{1}(\varOmega_{-}))^{\prime}\!, \\ & \; \theta \;&\; \longmapsto \,& \; b(\theta,\,\cdot\,;s) \end{aligned}$$ be the operators associated to the bilinear forms (3.9) and (3.12), respectively, and consider similarly the operators   $$\begin{aligned} \textrm{div}:\; &\mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (\mathbf H^{1}(\varOmega_{-}))^{\prime}\!, \\ &\;\mathbf u \;&\; \longmapsto \;&\; (\nabla\cdot\mathbf u,\,\cdot\,)_{\varOmega_{-}}, \\ \gamma_{n}:\; & \mathbf H^{1}(\varOmega_{-}) &\;\longrightarrow\;&\; (H^{1/2}(\varGamma))^{\prime}=H^{-1/2}(\varGamma), \\ &\;\mathbf u \;&\; \longmapsto \;&\; \langle\gamma^{-}\mathbf u\cdot\mathbf n,\,\cdot\,\rangle_{\varGamma}. \end{aligned}$$ The operator div is the distributional divergence operator followed by the canonical injection of $$L^{2}(\varOmega _{-})\equiv L^{2}(\varOmega _{-})^{\prime }$$ into the dual space of $$H^{1}(\varOmega _{-})$$. From (3.10), (3.12), (3.6) and (3.7), the nonlocal problem may be formulated as a system of operator equations: given data $$(d_{1}, d_{2}, d_{3}, d_{4}) \in X^{\prime },$$ find $$(\mathbf u, \theta , \phi , \lambda ) \in X $$ such that   \begin{align} \pmb{\mathcal{A}} \left(\begin{array}{c} \mathbf{u} \\ \theta \\ \phi\\ \lambda \\ \end{array}\right):= \left ( \begin{array}{cccc} \mathbf A_{s}& -\zeta~\textrm{div}^{\prime} & s ~\rho_{f}\; \gamma_{n}^{\prime} & 0 \\ s ~\eta ~\textrm{div} & B_{s} &0&0 \\ - s~ \gamma_{n} & 0 & W(s) & \!\!\! - \frac{1}{2} I + K^{\prime}(s) \\ 0 & 0 & \!\!\! \frac{1}{2}I - K(s) & V(s)\\ \end{array} \right ) \left(\begin{array}{c} \mathbf{u} \\ \theta\\ \phi\\ \lambda \\ \end{array}\right) = \left(\begin{array}{c} d_{1}\\ d_{2}\\ d_{3}\\ d_{4} \end{array}\right). \end{align} (3.13)In the above expression data is given by   \begin{align} d_{1} = -s~ \rho_{f}\; \gamma_{n}^{\prime}\gamma^{+}v^{\textrm{inc}}, \quad d_{2} = 0, \quad d_{3} = {\partial_{n}^{+} v^{\textrm{inc}}}, \quad d_{4} = 0. \end{align} (3.14)We have made use of the product spaces   \begin{align*} X := \,&{\mathbf H}^{1}( \varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^{1/2}(\varGamma) \times H^{-1/2}(\varGamma), \\ X^{\prime} :=\,& ({\mathbf H}^{1}(\varOmega_{-}))^{\prime} \times (H^{1}(\varOmega_{-}) )^{\prime} \times H^{-1/2}(\varGamma) \times H^{1/2}(\varGamma). \end{align*}Our aim is to show that equation (3.13) has a unique solution in X. We will do this in the next section. 4. Existence and uniqueness results Before considering the existence and uniqueness results, we first discuss the invertibility of the operator $$\pmb{\mathcal{A} }$$ in (3.13). We begin with the definitions of the following energy norms:   \begin{align} |\!|\!|\mathbf u|\!|\!|_{|s|, \varOmega_{-}}^{2} :=\, ( \mathbf C\boldsymbol\varepsilon({\mathbf u}), \boldsymbol{\varepsilon} (\bar{\mathbf u} ) )_{\varOmega_{-}} + \rho_{\varSigma} \| s \; \mathbf u \|^{2}_{\varOmega_{-}},\quad \mathbf u \in{\mathbf H}^{1}(\varOmega_{-}), \end{align} (4.1)  \begin{align} |\!|\!|\theta|\!|\!|^{2}_{|s|, \varOmega_{-}} :=\, \| \nabla \theta\|^{2}_{\varOmega_{-}} + \kappa^{-1} \| s^{1/2} \; \theta \|_{\varOmega_{-}}^{2},\quad \theta \in H^{1}(\varOmega_{-}), \end{align} (4.2)  \begin{align} |\!|\!|v|\!|\!|^{2}_{|s|, \varOmega_{+}} :=\, \| \nabla v\|^{2}_{\varOmega_{+}} + c^{-2} \| s \;v \|^{2}_{\varOmega_{+}},\quad v \in H^{1}(\varOmega_{+}). \end{align} (4.3)For the complex Laplace parameter s we will denote   $$ \sigma:= \textrm{Re}\;s, \quad \underline{\sigma}:= \min\{1,\sigma\}, $$and will make use of the following equivalence relations for the norms:   \begin{align} \underline{\sigma }|\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega_{-}} \leq |\!|\!|\mathbf{u}|\!|\!|_{|s|, \varOmega_{-}}\leq \frac{|s|}{\underline{\sigma}} |\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega_{-}}, \end{align} (4.4)  \begin{align} \sqrt{\underline{\sigma}}|\!|\!|\theta|\!|\!|_{1, \varOmega_{-}} \leq |\!|\!|\theta|\!|\!|_{|s|, \varOmega_{-}} \leq \sqrt{\frac{|s|}{\underline{\sigma} } }|\!|\!|\theta|\!|\!|_{1,\varOmega_{-}}, \end{align} (4.5)  \begin{align} \underline{\sigma } |\!|\!|v|\!|\!|_{1, \varOmega_{+}} \leq |\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}\leq \frac{|s|}{\underline{\sigma}} |\!|\!|v|\!|\!|_{1, \varOmega_{+}}, \end{align} (4.6)which can be obtained from the inequalities   $$ \min\{1, \sigma\} \leq \min\{1, |s|\}\quad \textrm{and} \quad \max\{1, |s|\} \min\{1, \sigma\} \leq |s|, \quad \forall\, s \in \mathbb{C}_{+}. $$We remark that the norms $$|\!|\!|\theta |\!|\!|_{1, \varOmega _{-}}$$ and $$ |\!|\!| v|\!|\!|_{1, \varOmega _{+}}$$ are equivalent to $$\| \theta \|_{H^{1}(\varOmega _{-})} $$ and $$ \|{ v}\|_{H^{1}(\varOmega _{+})}$$, respectively, and also the energy norm $$|\!|\!|\mathbf{u}|\!|\!|_{1, \varOmega _{-}}$$ is equivalent to the $$\mathbf{H}^{1}(\varOmega _{-})$$ norm of u by the second Korn inequality (Fichera, 1973). For the invertibility of $$\pmb{\mathcal{A} }$$, consider two invertible (for $$s\in \mathbb C_{+}$$) diagonal matrices   $$ \pmb{\mathcal Z_{1}}:= \left(\begin{array}{cccccc} \rho_{f}^{-1} & 0 & 0 & 0 \\ 0 & \eta^{-1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & s \end{array}\right), \qquad \pmb{\mathcal Z_{2}}:= \left(\begin{array}{cccccc} s^{-1} & 0 & 0 & 0 \\ 0 & \zeta^{-1}\rho_{f} & 0 & 0 \\ 0 & 0 & s^{-1} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$and note that   $$ \pmb{\mathcal Z_{1}}\pmb{\mathcal{A}}\pmb{\mathcal Z_{2}} =\pmb{\mathcal{B}}:=\left(\begin{array}{cccccc} (\rho_{f}\,s)^{-1}\mathbf A_{s}& -\textrm{div}^{\prime}& s \gamma_{n}^{\prime} & 0 \\ \textrm{div} & \frac{\rho_{f}}{\eta\zeta} B_{s} &0&0 \\ -\gamma_{n} & 0 & s^{-1} W(s) & - \frac{1}{2} I + K^{\prime}(s) \\ 0 & 0 & \frac{1}{2}I - K(s) &s V(s) \end{array}\right). $$By Banjai et al. (2015, Lemma 3.1), the block   $$ \left(\begin{array}{cccccc} s^{-1} W(s) & - \frac{1}{2} I + K^{\prime}(s) \\ \frac{1}{2}I - K(s) &s V(s) \end{array}\right) $$is coercive and it is therefore very simple to prove  $$ \textrm{Re}\Big( \big\langle \pmb{\mathcal{B}} (\mathbf v, \vartheta, \psi, \chi), \overline{ (\mathbf v, \vartheta, \psi, \chi) } \big\rangle \Big) \geq C_{1}\big(\zeta, \eta, \rho_{f} \big) \; C_{2}\big(|s|,\textrm{Re}\,s\big)\big \| (\mathbf v, \vartheta, \psi, \chi)\big\|^{2}_{X} $$for some positive $$C_{1}$$ and $$C_{2}$$ that are not relevant for this argument. This proves that $$\pmb{\mathcal{A}}$$ is invertible. Let us next study solutions of the problem   \begin{align} \pmb{\mathcal{A}} \left(\begin{array}{cccccc} \mathbf u \\ \theta \\ \phi \\ \lambda\end{array}\right) = \left(\begin{array}{cccccc} d_{1} \\ 0 \\ d_{3} \\ 0 \end{array}\right). \end{align} (4.7)Note that we have taken $$d_{2}=0$$ and $$d_{4}=0$$ as we did in the first occurrence of the operator $$\pmb{\mathcal{A}}$$ in (3.13). If we define   \begin{align} v:= \mathcal{D}(s) \phi - \mathcal{S}(s) \lambda \quad \textrm{in} \quad \mathbb{R}^{3} \setminus \varGamma, \end{align} (4.8)then $$v\in H^{1}(\mathbb{R}^{3} \setminus \varGamma )$$ is the solution of the transmission problem   \begin{align} - \varDelta v + (s/c)^{2} v = 0 \quad \textrm{in}\quad \mathbb{R}^{3} \setminus \varGamma \end{align} (4.9)satisfying the following jump relations across $$\varGamma $$:   $$ {[\kern-1.5pt[}\gamma v{]\kern-1.5pt]}:= \gamma^{+} v - \gamma^{-} v = \phi \in H^{1/2}\big(\varGamma\big), \quad {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} :=\partial_{n}^{+} v - \partial_{n}^{-} v = \lambda \in H^{-1/2}\big(\varGamma\big). $$The last two equations from (4.7) are then equivalent to   \begin{align} -s \;\gamma^{-}{\mathbf u }\cdot\mathbf n - \partial_{n}^{+} v = d_{3}\quad \textrm{on} \quad \varGamma, \end{align} (4.10)  \begin{align} - \; \gamma^{-} v = 0\quad \textrm{on} \quad \varGamma. \end{align} (4.11)The boundary conditions (4.11) and (4.9) imply that v = 0 in $$\varOmega _{-}$$ and therefore   \begin{align} {[\kern-1.5pt[}\gamma v{]\kern-1.5pt]} = \gamma^{+} v = \phi \quad \textrm{and} \quad {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} = \partial_{n}^{+} v = \lambda. \end{align} (4.12)We introduce the bilinear form in $$H^{1}(\varOmega _{+})$$,  $$ c_{\varOmega_{+}} \big(f, g; s\big):= \big(\nabla f,\nabla g\big)_{\varOmega_{+}} + \big(s/c\big)^{2}\big(f,g\big)_{\varOmega_{+}}, $$and the associated operator   $$\begin{aligned} C_{s, \varOmega_{+}}:&& \; H^{1}(\varOmega_{+}) \;& \longrightarrow\;& \left(H^{1}(\varOmega_{+})\right)^{\prime}\!,\\ &&\; f \;& \longmapsto \;& c_{\varOmega_{+}}(f,\;\cdot\;;s), \end{aligned}$$ By (4.9) and the transmission condition (4.10), we obtain   \begin{align} \big(C_{s, \varOmega_{+} }\, v,w\big)_{\varOmega_{+}} = &- \big\langle \partial^{+}_{n} v, \gamma^{+} w\big \rangle \nonumber\\ \,= &\, \big\langle d_{3}, \gamma^{+} w\big\rangle_{\varGamma} + \big\langle s ~\gamma^{-} \mathbf{u} \cdot \; \mathbf{n}, \gamma^{+} w\big \rangle_{\varGamma}. \end{align} (4.13)Therefore, the triple   $$ (\mathbf{u}, \theta, v) \in \pmb{\mathbb{H}} = {\mathbf H}^{1}(\varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^ 1(\varOmega_{+}) $$satisfies the system of operator equations   \begin{align} \left(\begin{array}{cccccc} \mathbf A_{s} & -\zeta \,\textrm{div}^{\prime} & s \rho_{f} \gamma_{n}^{\prime} \gamma^{+} \\ s\,\eta\,\textrm{div} & B_{s} & 0 \\ -s(\gamma^{+})^{\prime} \gamma_{n} & - & C_{s,\varOmega_{+}} \end{array}\right) \left(\begin{array}{cccccc} \mathbf u \\ \theta \\ v \end{array}\right) = \left(\begin{array}{cccccc} d_{1} \\ 0 \\ (\gamma^{+})^{\prime}d_{3} \end{array}\right). \end{align} (4.14)Returning to bilinear forms, these equations are equivalent to   \begin{align} \begin{aligned} a(\mathbf{u}, \mathbf v;s ) \! \! - \! \zeta( \theta,{\nabla\cdot} \; \mathbf v )_{\varOmega_{-}}\! \! + \!s \rho_{f} \langle \gamma^{+} v , \gamma^{-} \mathbf v \cdot\mathbf n\rangle_{\varGamma} = \,& (d_{1}, \mathbf v )_{\varOmega_{-}} \quad && \forall\, \mathbf v \in{\mathbf H}^{1}(\varOmega_{-}), \\ s\,\eta\,(\nabla\cdot \; \mathbf u, {\vartheta } )_{\varOmega_{-}} + b(\theta, \; \vartheta;s ) =\;& 0 \quad&& \forall \, \vartheta \in H^{1}(\varOmega_{-}), \\ - s \langle \gamma^{-} \mathbf{u} \cdot\mathbf n\;, \; {\gamma^{+} v} \rangle_{\varGamma}+ c_{\varOmega_{+}}( v,w;s) =\; & \langle d_{3}, {\gamma^{+} w} \rangle_{\varGamma} \quad&& \forall \ w \in H^{1}(\varOmega_{+}). \end{aligned} \end{align} (4.15)Note that the expression $$(d_{1}, \mathbf v )_{\varOmega _{-}}$$ in the right-hand side of (4.15) is a duality product and not an $$L^{2}$$ inner product. By construction, the variational problem (4.15) is equivalent to the operator equations (4.7) when we identify on the one hand $$\phi =\gamma ^{+} v$$ and $$\lambda =\partial _{\nu }^{+} v$$ and, on the other hand, $$v=\mathcal D(s)\phi -\mathcal S(s)\lambda $$. Theorem 4.1 The variational problem (4.15) has a unique solution $$(\mathbf{u}, \theta , v) \in \pmb{\mathbb{H}} $$. Moreover, the following estimate holds:   \begin{align} \left(\!|\!|\!|\mathbf{u}|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!|s^{-1/2}\theta|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!|v|\!|\!|^{2}_{|s|, \varOmega_{+}} \!\right)^{1/2} \leq c_{0} \; \frac{|s|}{\sigma{\underline{\sigma}}} ~ \big\|(d_{1}, 0, d_{3}, 0)\big\|_{X^{\prime}}, \end{align} (4.16)where $$c_{0}$$ is a constant depending only on the physical parameters $$ \rho _{f}, \zeta , \eta $$. Proof. Testing equations (4.15) with $$\mathbf v=\overline{s\,\mathbf u}$$, $$\vartheta =\zeta \,\eta ^{-1}\overline \theta $$, $$v=\rho _{f}\,\overline{s\, v}$$, and adding the result, it is simple to see that   \begin{align} \textrm{Re}\,\Big( \overline s\, &a(\mathbf u,\overline{\mathbf u};s) +\zeta\,\eta^{-1}\, b(\theta,\overline\theta;s) +\rho_{f} \overline s\,c_{\varOmega_{+}} (v,\overline v;s)\Big) \\ \nonumber &=\textrm{Re}\, \Big( \overline s \, (d_{1},\overline{\mathbf u})_{\varOmega_{-}} +\overline s\,\langle d_{3},\gamma^{+}\overline v\rangle_{\varGamma}\Big). \end{align} (4.17)A simple computation shows next that   \begin{align} \textrm{Re} \left( \overline s a(\mathbf u, \overline{\mathbf u};s)\right) =\; \sigma\, |\!|\!|\mathbf u|\!|\!|^{2}_{|s|,\varOmega_{-}}, \end{align} (4.18)  \begin{align} \textrm{Re}\, b(\theta,\overline{\theta};s) \geq\; \frac{\sigma}{|s|}~ |\!|\!|\theta|\!|\!|^{2}_{|s|, \varOmega_{-}}=\sigma |\!|\!|s^{-1/2}\theta|\!|\!|_{|s|,\varOmega_{-}}^{2}, \end{align} (4.19)  \begin{align} \textrm{Re} \left( \overline s\, c_{\varOmega_{+}} (v, \overline{v};s) \right) =\;\sigma\, |\!|\!|v|\!|\!|^{2}_{|s|,\varOmega_{+}}, \end{align} (4.20)which proves the well-posedness of the variational problem (4.15). Using these inequalities in (4.17) for the particular case $$d_{2}=0$$, and taking into account the inequalities (4.4) and (4.6) (relating energy norms to Sobolev norms), we can easily prove the result. Theorem 4.2 If   $$ (d_{1},d_{3})\in (\mathbf H^{1}(\varOmega_{-}))^{\prime}\times H^{-1/2}(\varGamma), $$then for the unique solution of (4.7) we have the estimates   \begin{align} |\!|\!|\mathbf u|\!|\!|_{1,\varOmega_{-}}+\big\|\phi\big\|_{H^{1/2}(\varGamma)} \leq c_{1} \frac{|s|}{\sigma\,\underline\sigma^{2}} \big\| \big(d_{1},0,d_{3},0\big)\big\|_{X^{\prime}}, \end{align} (4.21)  \begin{align} |\!|\!|\theta|\!|\!|_{1,\varOmega_{-}}+\big\|\lambda\big\|_{H^{-1/2}(\varGamma)} \leq c_{2} \frac{|s|^{3/2}}{\sigma\,\underline\sigma^{3/2}} \big\| \big(d_{1},0,d_{3},0\big)\big\|_{X^{\prime}}, \end{align} (4.22)where $$c_{1}$$ and $$c_{2}$$ are constants independent of s. Proof. The bounds for u and $$\theta $$ follow from Theorem 4.1 and (4.4)–(4.5). Since, as we pointed out in (4.12), we have   $$ \gamma^{+}v={[\kern-1.5pt[}\gamma v{]\kern-1.5pt]} = \phi \in H^{1/2}(\varGamma), \quad \partial_{n}^{+} v= {[\kern-1.5pt[}\partial_{n} v{]\kern-1.5pt]} =\lambda \in H^{-1/2}(\varGamma), $$we thus have the estimate (see, e.g., Hsiao et al., 2013)   \begin{align} \big\|\phi\big \|_{H^{1/2}(\varGamma)} = \big\|\gamma^{+} v\big\|_{H^{1/2}(\varGamma)} \leq C |\!|\!|v|\!|\!|_{1, \varOmega_{+}} \leq C \frac{1}{{\underline{\sigma}}} |\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}, \end{align} (4.23)therefore the bound for $$\phi $$ follows from Theorem 4.1 again. Similarly, an application of Bamberger & Ha-Duong’s (1986) optimal lifting leads to the estimate   \begin{align} \big\|\lambda\big\|_{H^{-1/2}(\varGamma)} =\big\|\partial^{+}_{n} v\big \|_{H^{-1/2}(\varGamma)} \leq c_{2} \big(|s|/\underline\sigma\big)^{1/2}|\!|\!|v|\!|\!|_{|s|, \varOmega_{+}}. \end{align} (4.24)A detailed proof of this inequality can be found in Sayas (2016, Proposition 2.5.2). Therefore, Theorem 4.1 provides the bound for $$\lambda $$. We remark that in view of (3.2), we see that $$\mathbf{u}, \theta $$ and v are solutions of the system   \begin{align} \left(\begin{array}{c} \mathbf{u} \\ \theta\\ v\\ \end{array}\right) = \left ( \begin{array}{cccc} 1& 0 & 0 & 0\\ 0 & 1& 0 & 0 \\ 0 & 0 &\mathcal{D}(s) & - \mathcal{S}(s)\\ \end{array} \right ) {\pmb{\mathcal{A}}}^{-1} \left(\begin{array}{c} d_{1} \\ 0\\ d_{3}\\ 0\\ \end{array}\right). \end{align} (4.25)With the properties of solutions available in the transformed domains, we are now in a position to estimate the corresponding properties of solutions in the time domain based on some inversion theorems of the Laplace transform (see Lubich, 1994 for an early version and Sayas, 2016, Proposition 3.2.2 for the result as we use it). The crucial result described in Proposition 4.3 below is employed to retrieve time-domain estimates from those obtained in the Laplace domain. Before presenting the aforementioned result we must introduce some notation. For Banach spaces X and Y, let $$\mathcal{B}(X, Y)$$ denote the set of bounded linear operators from X to Y. We say that an analytic function $$A : \mathbb{C}_{+} \rightarrow \mathcal{B}(X, Y)$$ is an element of the class of symbols $$\mathbf{Sym} (\mu , \mathcal{B}(X, Y))$$ if there exist $$\mu \in \mathbb{R}$$ and $$m\geq 0$$ such that   $$ \big\|A(s)\big\|_{X,Y} \leq C_{A}(\textrm{Re}~s) |s|^{\mu} \quad \textrm{for}\quad s \in \mathbb{C}_{+}, $$where $$C_{A} : (0, \infty ) \rightarrow (0, \infty ) $$ is a nonincreasing function such that   $$ C_{A}(\sigma) \leq \frac{ c}{\sigma^{m}} \quad \forall \, \sigma \in ( 0, 1].$$In order to make the statement of the time-domain estimates more compact, we will make use of the regularity spaces   \begin{align} W^{k}_{+}( \mathcal{H}):= \Big\{ w \in \mathcal{C}^{k-1}(\mathbb{R}; \mathcal{H}) : w ~\equiv 0~ \textrm{in} ~(-\infty,0), w^{(k)} \in L^{1} (\mathbb{R}; \mathcal{H})\Big \}, \end{align} (4.26)where $$\mathcal{H}$$ denotes a Banach space. Proposition 4.3 (Laliena & Sayas, 2009; Sayas, 2016). Let $$A = \mathcal{L}\{a\} \in \mathbf{Sym}\big (\mu , \mathcal{B}\big (X, Y\big )\big )$$ with $$\mu \geq 0$$ and let   $$ k:=\big\lfloor \mu +2\big \rfloor, \quad \varepsilon := k - (\mu +1) \in (0, 1]. $$If $$ g \in W_{+}^{k}(\mathbb R,X)$$, then $$a* g \in \mathcal{C}(\mathbb{R}, Y)$$ is causal and   $$ \big\|\big (a*g\big)\big(t\big)\big \|_{Y} \leq 2^{\mu+1} C_{\varepsilon} \big(t\big) C_{A} \big(t^{-1}\big) {\int_{0}^{1}} \big\|\big(\mathcal{P}_{k}g\big)\big(\tau\big)\big \|_{X} \; \textrm{d}\tau, $$where   $$ C_{\varepsilon} \big(t\big) := \frac{t^{\varepsilon} }{\pi \varepsilon } \qquad\textrm{and}\qquad \big(\mathcal{P}_{k}g\big) \big(t\big) = \sum_{\ell=0}^{k} {{k}\choose{\ell}} g^{(\ell)}(t). $$ Theorem 4.4 If $$\mathbf{D} := \mathcal{L}^{-1}\{ (d_{1}, 0, d_{3}, 0)\}\in W_{+}^{3}(\mathbb R, X^{\prime })$$, then   \begin{align*} \big(\mathbf{U}, \varTheta, \mathcal{L}^{-1} \big\{\phi\big\}, \mathcal{L}^{-1}\big\{ \lambda\big \}\big) & \in \mathcal{C}\big(\big[0,T\big],X\big),\\ \big(\mathbf{U}, \varTheta, V\big) \in C\big( \big[0, T\big], \pmb{\mathbb{H}} \big) \end{align*}and we have the bounds   \begin{align*} \big\| \mathbf U(t)\big\|_{\mathbf H^{1}(\varOmega_{-})} + \big\|\mathcal L^{-1}\big\{\phi\big\}(t)\big\|_{H^{1/2}(\varGamma)} +\big\| V(t)\big\|_{H^{1}(\varOmega_{+})} & \leq c t^{2} \max\big\{1,t^{2}\big\} {\int_{0}^{t}} \big\| (\mathcal{P}_{3}\mathbf D) (\tau)\big\|_{X^{\prime}}\; \textrm{d} \tau,\\ \big\| \varTheta(t)\big\|_{H^{1}(\varOmega_{-})} + \big\|\mathcal L^{-1}\big\{\lambda\big\}\big\|_{H^{-1/2}(\varGamma)} &\leq c t^{\frac32} \max\big\{1,t^{\frac32}\big\} {\int_{0}^{t}} \big\| (\mathcal{P}_{3}\mathbf D) (\tau)\big\|_{X^{\prime}}\; \textrm{d} \tau. \end{align*} Proof. For the bounds for U(t) and $$\mathcal L^{-1}\big \{\phi \big \}(t)$$, use (4.21) and Proposition 4.3 with $$\mu =1$$, k = 3, $$\varepsilon =1$$. Note that a bound for V (t) follows from the same proposition now using Theorem 4.1 and (4.6). The bounds for $$\varTheta (t)$$ and $$\mathcal L^{-1}\big \{\lambda \big \}(t)$$ follow from (4.22) and Proposition 4.3 with $$\mu =3/2$$, k = 3, $$\varepsilon =1/2$$. 5. Semidiscrete error estimates In this section we discuss the results concerning the discretization of (3.13). We begin with the Galerkin semidiscretization in the space of the system of equations. Let   $$ {\mathbf V}_{h} \subset \mathbf H^{1}(\varOmega_{-}), \quad W_{h} \subset H^{1}(\varOmega_{-}), \quad Y_{h} \subset H^{1/2}(\varGamma), \quad X_{h} \subset H^{-1/2} (\varGamma) $$be families of finite-dimensional subspaces. We say $$ \left (\mathbf u^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\right ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$ is a Galerkin solution of (4.7) if it satisfies the Galerkin equations   \begin{align} \Big(\pmb{\mathcal{A}} \big(\mathbf u^{h}, \theta^{h}, \phi^{h}, \lambda^{h}\big)^{\textrm{T}},\big(\mathbf v,\vartheta,\psi,\eta\big)^{\textrm{T}}\Big) = \Big(\big(d_{1},d_{2},d_{3},0\big)^{\textrm{T}},\big(\mathbf v,\vartheta,\psi,\eta\big)^{\textrm{T}}\Big) \end{align} (5.1)for all $$(\mathbf v, \vartheta , \psi , \eta ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$. Solutions of Galerkin equations of (5.1) can be established in the same manner as the exact solutions of the system (4.7). We will not repeat the process and we consider only the error estimates here. We note that if $$\big ({\mathbf u}^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\big ) \in{\mathbf V}_{h} \times W_{h} \times Y_{h} \times X_{h} $$ is a Galerkin solution of (5.1), then   \begin{align} v^{\,h} := \mathcal{D}(s) \phi^{h} - \mathcal{S}(s) \lambda^{h} \; \in H^{1}\big(\mathbb R^{3}\setminus\varGamma\big) \end{align} (5.2)satisfies   \begin{align} -\varDelta v^{h} + (s/c)^{2}v^{h} = \; & 0 \qquad \mbox{in } \quad \mathbb{R}^{3} \setminus \varGamma,\\ \nonumber {[\kern-1.5pt[}\gamma v^{h}{]\kern-1.5pt]} = \;& \phi^{h}\in Y_{h} \subset H^{1/2}(\varGamma), \\ \nonumber {[\kern-1.5pt[}\partial v^{h}{]\kern-1.5pt]} = \;& \lambda^{h} \in X_{h} \subset H^{-1/2}(\varGamma). \end{align} (5.3)Now set   $$ V_{h} := \big\{ w \in H^{1}\big(\mathbb{R}^{3} \setminus \varGamma \big) : {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]} \in Y_{h}, \gamma^{-}w \in X^{\circ}_{h} \big\}. $$Here and in the sequel, the upper script $$^{\circ }$$ will be used to denote the annihilator (or polar set) of a subspace of a Banach space. In this particular case,   $$ X_{h}^{\circ} := \big\{ w \in X^{\prime}_{h} : \big\langle v, w\big \rangle = 0 \quad \forall\, v \in X_{h} \big\}. $$Applying Green’s formula to (5.3), we obtain for $$w \in V_{h}$$,   \begin{align} \big\langle \partial_{n}^{+} v^{h}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} = - c_{\mathbb R^{3}\setminus\varGamma}\;\big(v^{h}, w\; ; s\big) - \big\langle{[\kern-1.5pt[}\partial_{n} v^{h}{]\kern-1.5pt]},\gamma^{-} w\big\rangle_{\varGamma}. \end{align} (5.4)As a consequence, we see that $$ \big ( \mathbf u^{h}, \theta ^{h}, v^{h}\big ) \in{\mathbf V}_{h} \times W_{h} \times V_{h} $$ satisfies the variational equations   \begin{align} a\big(\mathbf{u}^{h}, \mathbf v; s \big) - \zeta \big( \theta^{h}, \nabla\cdot \; \mathbf v \big)_{\varOmega_{-}}\! + s\rho_{f}\big\langle {[\kern-1.5pt[}\gamma v^{h}{]\kern-1.5pt]}, \gamma^{-} \mathbf v \cdot \mathbf n\big\rangle_{\varGamma} = \; & \big(d_{1}, \mathbf v\big)_{\varOmega_{-}} &\quad& \forall\, \mathbf v \in{\mathbf V}^{h}, \nonumber \\ s\, \eta\, \big( \nabla\cdot\; \mathbf u^{h}, \vartheta\big)_{\varOmega_{-}}\! + b\big(\theta^{h}, \vartheta; s\big) =\;& \big(d_{2}, \vartheta\big)_{\varOmega_{-}}&\quad& \forall\, \vartheta \in W_{h}, \\ - s\big \langle \gamma^{-} \mathbf{u}^{h} \cdot \mathbf{n}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} + c_{\mathbb R^{3}\setminus \varGamma} \big( v^{h}, w; s\big ) =\;& \big\langle d_{3}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle_{\varGamma} &\quad& \forall\ w ~ \in V_{h}. \nonumber \end{align} (5.5)For the error estimate, we need to compare $$\big (\mathbf u^{h}, \theta ^{h}, v^{h}\big )$$ with the exact solution $$\big (\mathbf u, \theta , v\big )$$ of the transmission problem. The exact solution $$\big (\mathbf u, \theta , v\big ) \in \mathbf H^{1}\big (\varOmega _{-}\big ) \times H^{1}\big (\varOmega _{-}\big ) \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big )$$ satisfies the variational equation (4.15), understanding that v ≡ 0 in $$\varOmega _{-}$$. This implies that   \begin{align} a\big(\mathbf{u}^{h}\!- \mathbf u, \mathbf v; s \big) - \zeta\big( \theta^{h} \!- \theta, \nabla\cdot \; \mathbf v \big)_{\varOmega_{-}} \! + s\; \rho_{f} \big\langle[\![\gamma \big(v^{h} \!-v\big)]\!] , \gamma^{-} \mathbf v\cdot \mathbf n \big\rangle_{\varGamma} &\; = 0 \quad \forall\, \mathbf v \in{\mathbf V}^{h}, \nonumber \\ \frac{s}{|s|} \zeta\left(\nabla\cdot\big(\mathbf u^{h} \!- \mathbf u\big), \vartheta \right)_{\varOmega_{-}}\! + \frac{\zeta}{\eta} \frac{1}{|s|} b\big(\theta^{h}\!- \theta, \vartheta; s\big) &\; = 0 \quad \forall\, \vartheta \in W_{h}, \\ \nonumber - s\rho_{f} \big\langle \gamma^{-} \big(\mathbf{u}^{h}\!- \mathbf u\big)\cdot\mathbf n, [\![\gamma w]\!]\big\rangle_{\varGamma} + \rho_{f} c_{\mathbb R^{3}\setminus \varGamma} \big( v^{h}\! - v, w; s \big) \quad & & & \\ \nonumber + \rho_{f} \big\langle [\![\partial_{n}\big(v^{h}\! - v\big)]\!], \gamma^{-} w \big\rangle_{\varGamma} &\; = 0 \quad \forall\ w \in V_{h}. \nonumber \end{align} (5.6)Note that we have multiplied the second and third equations by $$\eta /(|s|\zeta )$$ and $$\rho _{f}$$, respectively, mimicking what we did in the proof of Theorem 4.1. The significance of (5.6) is that it indicates that the Galerkin solutions are the best possible approximations of the exact solution in the finite-dimensional subspaces with respect to the inner products defined by the underlying bilinear forms. However, it is worth emphasizing that the errors $$ \big (\mathbf u^{h} -\mathbf u \big )$$ and $$\big (\theta ^{h} - \theta \big )$$ do not belong to the discrete function spaces. In order to justify (5.6) as a proper variational formulation, we will make use of the spaces of constants and of infinitesimal rigid motions   \begin{align} \mathfrak{R}_{\mathbf u} := \;\big\{ \mathbf m \in \mathbf H^{1}(\varOmega_{-}): ( \mathbf C \boldsymbol\varepsilon (\mathbf m), \boldsymbol{\varepsilon}(\mathbf m))_{\varOmega_{-}} = 0 \big\}, \end{align} (5.7)  \begin{align} {\mathfrak{R}}_{\theta} := \; \big\{ m \in H^{1}(\varOmega_{-}): ( \nabla m, \nabla m)_{\varOmega_{-}} = 0 \big\}, \end{align} (5.8)which in what follows will always be assumed to be contained in the discrete subspaces $$\mathbf V_{h}$$ and $$W_{h}$$, respectively. We now define the elliptic projection on displacement fields   \begin{align} \begin{aligned} \mathbf P_{h}: \mathbf H^{1}(\varOmega_{-}) \longrightarrow \;&\mathbf V_{h} \subset \mathbf H^{1}(\varOmega_{-}), & \\ \big(\mathbf C\boldsymbol\varepsilon(\mathbf P_{h} \mathbf u), \boldsymbol{\varepsilon} \big(\mathbf v^{h}\big)\big)_{\varOmega_{-}} = \;& \big(\mathbf C\boldsymbol\varepsilon(\mathbf u), \boldsymbol{\varepsilon} \big(\mathbf v^{h} \big) \big)_{\varOmega_{-}} \qquad & \forall\, \mathbf v^{h}\in \mathbf V_{h},\\ (\mathbf P_{h} \mathbf u, \mathbf m)_{\varOmega_{-}} =\;& (\mathbf u, \mathbf m)_{\varOmega_{-}} \qquad & \forall\ \mathbf m\in \mathfrak R_{\mathbf u}. \end{aligned} \end{align} (5.9)This projection is well defined thanks to Korn’s second inequality and can be alternatively introduced as the orthogonal projection onto $$\mathbf V_{h}$$ with respect to the nonstandard (but equivalent) inner product in $$\mathbf H^{1}(\varOmega),$$  $$ (\mathbf C\boldsymbol\varepsilon(\mathbf u),\boldsymbol\varepsilon(\mathbf v))_{\varOmega_{-}}+\big(\mathbf P_{\mathfrak R_{\mathbf u}}\mathbf u,\mathbf P_{\mathfrak R_{\mathbf u}}\mathbf v\big)_{\varOmega_{-}}, $$where $$\mathbf P_{\mathfrak R_{\mathbf u}}$$ is the $$\mathbf L^{2}(\varOmega _{-})$$ orthogonal projection onto $$\mathfrak R_{\mathbf u}$$. This implies that the approximation error $$\|\mathbf u-\mathbf P_{h}\mathbf u\|_{1,\varOmega _{-}}$$ is equivalent to the best approximation error in $$\mathbf H^{1}(\varOmega _{-})$$ by elements of $$\mathbf V_{h}$$. Similarly, we can introduce a projection on the discrete scalar fields   \begin{align} \begin{aligned} Q_{h}: H^{1}(\varOmega_{-}) \longrightarrow \;& W_{h} \subset H^{1}(\varOmega_{-}), & \\ \big(\nabla (Q_{h} \theta), \nabla \vartheta^{h}\big)_{\varOmega_{-}} = \;& \big(\nabla \theta, \nabla\vartheta^{h} \big)_{\varOmega_{-}} \qquad & \forall\,\vartheta^{h}\in W_{h},\\ (Q_{h} \theta, m)_{\varOmega_{-}} = \;&(\theta, m)_{\varOmega_{-}}\qquad &\forall\ m\in \mathfrak R_{\theta}. \end{aligned} \end{align} (5.10)Note that the reason to introduce these projections is to avoid having additional mass terms arising from the full Sobolev norm in the associated error equations. In terms of the elliptic projection $$\mathbf P_{h}$$, we can define   $$ \mathbf e_{\mathbf u}^{h} := \mathbf u^{h} - \mathbf P_{h}\mathbf u, \qquad \mathbf r_{\mathbf u}^{h} := \mathbf P_{h} \mathbf u - \mathbf u, $$so that we can decompose   $$ \mathbf u^{h} - \mathbf u = \big(\mathbf u^{h} - \mathbf P_{h} \mathbf u \big) + (\mathbf P_{h} \mathbf u - \mathbf u) = \mathbf e_{\mathbf u}^{h} + \mathbf r_{\mathbf u}^{h}. $$As a consequence,   $$ a \big(\mathbf u^{h} - \mathbf u, \mathbf v^{h} ; s\big) = a \big(\mathbf e_{\mathbf u}^{h} \;, \mathbf v^{h}; s\big) + s^{2} \rho_{\varSigma} \big(\mathbf r_{\mathbf u}^{h}, \mathbf v^{h}\big)_{\varOmega_{-}}. $$We may decompose the error $$\theta ^{h} - \theta $$ in a similar manner by letting   $$ e^{h}_{\theta}:= \theta^{h} - Q_{h}\theta, \qquad r^{h}_{\theta} := Q_{h}\theta - \theta, $$so that $$\theta ^{h}-\theta = e^{h}_{\theta } + r^{h}_{\theta }$$ and therefore   $$ b \big(\theta^{h} -\theta, \vartheta^{h}; s \big) = b \big(e^{h}_{\theta}, \vartheta^{h}; s\big) + (s/\kappa) (r^{h}_{\theta}, \vartheta^{h})_{\varOmega_{-}}. $$Finally, we define   $$ {e_{v}^{h}}:= \mathcal D(s) (\phi^{h}-\phi) - \mathcal S(s)(\lambda^{h}-\lambda) \qquad \textrm{ in }\; \mathbb R^{3}\setminus\varGamma. $$This leads to the variational formulation for the error functions $$\big (\mathbf e_{\mathbf u}^{h}, e^{h}_{\theta }, {e^{h}_{v}}\big ) \in \mathbf V_{h} \times W_{h} \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big )$$. Theorem 5.1 The error functions $$ \big (\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta }, {e^{h}_{v}}\big ) \in \mathbf V_{h} \times W_{h} \times H^{1}\big (\mathbb R^{3} \setminus \varGamma \big ) $$ satisfy the variational formulation   \begin{align*} \big(\gamma^{-} {e^{h}_{u}},\; {[\kern-1.5pt[}\gamma{e^{h}_{u}}{]\kern-1.5pt]}\; +\phi,\; {[\kern-1.5pt[}\partial_{n} {e^{h}_{v}}{]\kern-1.5pt]} + \lambda\big) \in \; & X^{\circ}_{h} \times Y_{h}\times X_{h}, & \\ \mathcal{A}((\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}), ( \mathbf v, \vartheta, w) ;s ) =\; & \ell((\mathbf v, \vartheta, w) ; s) \quad\forall\, (\mathbf v, \vartheta, w) \in \mathbf V_{h} \times W_{h} \times V_{h}. \end{align*}Where the bilinear form $$\mathcal A$$ is defined by   \begin{align} \mathcal{A}\big(\big(\mathbf{e}^{h}_{\mathbf{u}}, e^{h}_{\theta}, {e^{h}_{v}}\big), ( \mathbf{v}, \vartheta, w) ;s \big) :=\;& a \big(\mathbf{e}^{h}_{\mathbf{u}}, \mathbf{v}; s \big) + \frac{s}{|s|} \zeta\big({\nabla\cdot} ~ \mathbf{e}^{h}_{\mathbf{u}}, \vartheta\big)_{\varOmega_{-}} - s\rho_{f}\big\langle \gamma^{-} \mathbf{e}^{h}_{\mathbf{u}} \;, [\![\gamma w]\!]\mathbf{n}\big\rangle_{\varGamma} \nonumber \\ & + \frac{\zeta}{\eta} \frac{1}{|s|} b\; \big(e^{h}_{\theta}, \vartheta; s\big) - \zeta \big(e^{h}_{ \theta}, {\nabla\cdot}\; \mathbf{v}\big)_{\varOmega_{-}} \nonumber \\ & +\rho_{f} c_{\mathbb{R}^{3}\setminus \varGamma} \; \big( {e^{h}_{v}}, w; s \big) + s \rho_{f}\big\langle [\![\gamma{e^{h}_{v}}]\!] \; \mathbf{n}, \gamma^{-} \mathbf{v} \big\rangle_{\varGamma}, \end{align} (5.11)and the functional ℓ is given by   \begin{align} \nonumber \ell ((\mathbf v, \vartheta, w ) ; s) :=\; & - s^{2} \rho_{\varSigma}\big(\mathbf r^{h}_{\mathbf u}, \mathbf v \big)_{\varOmega_{-}} + \zeta\big(r^{h}_{\theta},{\nabla\cdot}~\mathbf v\big)_{\varOmega_{-}} -\frac{s}{|s|} \zeta \big({\nabla\cdot}~\mathbf r^{h}_{\mathbf u}, \vartheta\big)_{\varOmega_{-}}\\ & -\frac{\zeta}{\eta} \frac{s}{|s|}\frac{1}{\kappa}\big( r^{h}_{\theta}, \vartheta\big)_{\varOmega_{-}} +s \rho_{f} \big\langle \gamma^{-} \mathbf r^{h}_{ \mathbf u}, [\![\gamma w]\!] \mathbf n \big\rangle_{\varGamma} + \rho_{f} \langle \lambda, \gamma^{-}w \rangle_{\varGamma}. \end{align} (5.12) Proof. The bilinear form $$\mathcal{A}$$ follows easily from the left-hand side of (5.5) replacing $$\big (\mathbf{u}^{h} - \mathbf{u}, \theta _{h} - \theta , v^{h} - v\big ) $$ by $$\big ( \mathbf{e}^{h}_{\mathbf{u}} +\mathbf{r}^{h}_{\mathbf{u}}, e^{h}_{\theta } + r^{h}_{\theta }, {e^{h}_{v}}\big ) $$ and taking special care of the term $$e^{h}_{\theta }$$. From Green’s formula (5.4), we have   $$ \big\langle \partial_{n}^{+} {e^{h}_{v}}\;, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\big\rangle = - c_{\mathbb R^{3}\setminus \varGamma}\;\big({e^{h}_{v}}, w\; ; s\big) - \big\langle{[\kern-1.5pt[}\partial_{n} {e_{v}^{h}}{]\kern-1.5pt]}, \gamma^{-} w~ \big\rangle. $$But equations (4.10) and (4.11) imply   $$ -s \mathbf n\cdot \big(e^{h}_{\mathbf u} + \mathbf r^{h}_{\mathbf u}\big) -\partial_{n}^{+} {e^{h}_{u}} \in Y_{h}^{\circ} \quad \textrm{and} \quad \gamma^{-} e^{h} \in X^{\circ}_{h}. $$Hence,   $$ - s\rho_{f} \big\langle \gamma^{-} \mathbf e^{h}_{\mathbf u}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]}\mathbf{n} \big\rangle_{\varGamma} + \rho_{f} c_{\mathbb R^{3}\setminus \varGamma} \; \big( {e^{h}_{v}}, w; s \big)= s \rho_{f}\big\langle \gamma^{-} \mathbf r^{h}_{ \mathbf u}, {[\kern-1.5pt[}\gamma w{]\kern-1.5pt]} \mathbf n\big\rangle_{\varGamma} -\rho_{f}\big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}, \gamma^{-} w \big\rangle_{\varGamma}. $$We can rewrite the last term on the right-hand side as   $$ -\rho_{f} \big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}, \gamma^{-} w \big\rangle_{\varGamma}= -\rho_{f} \big\langle {[\kern-1.5pt[}\partial{e^{h}_{v}}{]\kern-1.5pt]}+ \lambda - \lambda, \gamma^{-} w \big\rangle_{\varGamma} = \rho_{f} \langle \lambda, \gamma^{-} w \rangle_{\varGamma}, $$where we have used that $$\lambda ^{h} = [\![\partial _{n} {e^{h}_{v}}]\!] + \lambda \in X_{h} $$, and $$ \gamma ^{-} w \in X^{\circ }_{h} $$ but $$ \lambda \not \in X_{h}$$. This completes the proof. Following arguments similar to those employed in the proof of Theorem 4.1, we can obtain the error estimate. In the following, for simplicity, let   $$ |\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}}\big)|\!|\!|^{2}_{|s|} := |\!|\!| \mathbf e^{h}_{\mathbf u}|\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!| e^{h}_{\theta} |\!|\!|^{2}_{|s|, \varOmega_{-}} + |\!|\!| {e^{h}_{v}} |\!|\!|^{2}_{|s|, \mathbb{R}^{3} \setminus \varGamma} . $$ Theorem 5.2 For $$(\mathbf u, \theta , \phi , \lambda ) \in \mathbf H^{1}(\varOmega _{-}) \times H^{1}(\varOmega _{-}) \times H^{1/2}(\varGamma ) \times H^{-1/2} (\varGamma ) $$, there holds the error estimate   \begin{align} |\!|\!|\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)|\!|\!|_{|s|} \leq C \frac{|s|^{2}}{\sigma{\underline{\sigma}}^{3}} \Big(\| \lambda \|_{-1/2,\; \varGamma}^{2} + \|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\Big), \end{align} (5.13)where the constant C depends only on the geometry and physical parameters. Proof. It is easy to see from the definition of the bilinear form $$\mathcal{A}$$ in (5.11) that there is a constant C depending only on the geometry and physical parameters such that   \begin{align} |\mathcal{A} \big(\big(e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big), (\mathbf v, \vartheta, w); s\big)| \leq C \frac{|s|}{\sigma \underline{\sigma}} |\!|\!|\big(e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)|\!|\!|_{1}~ |\!|\!|(\mathbf v, \vartheta, w)|\!|\!|_{|s|} . \end{align} (5.14)We also need the estimate for the functional,   \begin{align} |\ell((\mathbf v, \vartheta, w);s)| \leq \frac{C}{\underline{\sigma}} \Big(\| \lambda \|_{-1/2,\; \varGamma} + \|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}}+ \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\Big)|\!|\!|(\mathbf v, \vartheta, w)|\!|\!|_{|s|}. \end{align} (5.15)For $$\phi \in H^{1/2} (\varGamma )$$, we pick a lifting $$v_{\phi } \in H^{1}(\mathbb{R}^{3} \setminus \varGamma )$$ such that $$\gamma ^{+} v_{\phi } = \phi , \gamma ^{-}v_{\phi } = 0$$. Thus,   $$ \| v_{\phi} \|_{1, \mathbb{R}^{3} \setminus \varGamma} \le C \| \phi \|_{1/2, \varGamma}. $$Since $$\big (\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta }, {e^{h}_{v}} + v_{\phi } \big ) \in \mathbf V_{h} \times W_{h} \times V_{h}$$, it follows from equations (4.18)–(4.20) that   \begin{align*} \frac{\sigma \underline{\sigma} }{|s|^{2}}|\!|\!|\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big)|\!|\!|^{2}_{|s|} \leq \;& \big |\mathcal{A}\big(\big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big), \big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big); s\big) \big |\\ =\; & \big | \ell \big( \big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big) ; s\big) + \mathcal{A} \big((\mathbf 0, 0, v_{\phi}), \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}} + v_{\phi} \big) ;s ) \big | \\ \leq\; & \frac{C}{\underline\sigma}|\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}}+v_{\phi}\big)|\!|\!|_{|s|} \Big(\|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}}\!\!\\ &\;\; \qquad \qquad \qquad \qquad\qquad + \|r^{h}_{\theta}\|_{1, \varOmega_{-}}\! +\| \lambda \|_{-1/2,\; \varGamma} + \!\frac{|s|}{\sigma} \| v_{\phi} \|_{\mathbb{R}^{3} \setminus \varGamma} \Big) \\ \leq\;& \frac{C}{\underline{\sigma}^{2}}|\!|\!|\big(\mathbf e^{h}_{\mathbf u},\; e^{h}_{\theta}, \; {e^{h}_{v}} + v_{\phi}\big)|\!|\!|_{|s|} \Big(\|s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}}\!\! + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} \\ &\;\; \qquad \qquad \qquad \qquad\qquad\ +\|r^{h}_{\theta}\|_{1, \varOmega_{-}} + \| \lambda \|_{-1/2,\; \varGamma} + \|s \phi\|_{1/2, \varGamma}\Big), \end{align*}and the result follows from this relation and the observation that   $$ |\!|\!|(\mathbf 0,0,v_{\phi})|\!|\!|\leq \frac{C}{\underline\sigma}\|s\phi\|_{1/2,\varGamma} .$$ We are now in a position to establish the following result. Corollary 5.3 If $$(\mathbf u, \theta , \phi , \lambda ) \in \mathbf H^{1}(\varOmega _{-}) \times H^{1}(\varOmega _{-}) \times H^{1/2}(\varGamma ) \times H^{-1/2}(\varGamma ) $$ is the unique solution of the problem (4.7) and $$\big (\mathbf u^{h}, \theta ^{h}, \phi ^{h}, \lambda ^{h}\big ) $$ is its Galerkin approximation (5.1), then we have the estimates   \begin{align} \nonumber |\!|\!|(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}})|\!|\!|_{1} + \| \phi^{h} - \phi \|_{ 1/2, \varGamma} \leq\;& C \frac{|s|^{2}}{\sigma \underline{\sigma}^{4}} \Big( \| s \phi \|_{1/2, \varGamma} + \| \lambda \|_{-1/2,\; \varGamma}\\ &\qquad\quad\ + \| s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}} \Big), \end{align} (5.16a)  \begin{align} \nonumber \|\lambda^{h} - \lambda \|_{-1/2, \varGamma} \leq\; & C \frac{|s|^{5/2}}{\sigma\underline{\sigma}^{7/2}} \Big(\| s \phi \|_{1/2, \varGamma} + \| \lambda \|_{-1/2,\; \varGamma} \\ &\qquad\qquad + \| s^{2} \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|s \mathbf r^{h}_{\mathbf u} \|_{1, \varOmega_{-}} + \|\mathbf r^{h}_{\mathbf u}\|_{1, \varOmega_{-}} + \|r^{h}_{\theta}\|_{1, \varOmega_{-}} \Big). \end{align} (5.16b) Regarding the proof of this result, we would like to point out only that the estimate (5.16a) follows from a combined application of (5.13) in Theorem 5.2 and (4.23) by making use of the jump condition $$\phi ^{h} -\phi =[\![\gamma{e^{h}_{v}}]\!]$$. On the other hand, in order to establish the estimate (5.16b), one has to recall that $$\lambda -\lambda ^{h} =[\![\partial _{\nu } {e^{h}_{v}}]\!]$$ and therefore an application of (4.24) combined with (5.13) yields the desired inequality. Corollary 5.3 has the awkward aspect of being an error estimate where part of the right-hand side (the terms $$\|s\phi \|_{1/2,\varGamma }$$ and $$\|\lambda \|_{-1/2,\varGamma }$$) does not converge to zero. We now clarify why this is not so. Consider the best approximation operators   $$ \varPi_{X_{h}} : H^{-1/2} (\varGamma) \mapsto X_{h} \quad \textrm{and} \quad \varPi_{Y_{h}} : H^{1/2} (\varGamma)\mapsto Y_{h}. $$If we create data for the problem so that the exact solution is $$\big (\mathbf 0,0,\varPi _{Y_{h}}\phi ,\varPi _{X_{h}}\lambda \big )$$, then the associated numerical solution will be the exact solution and there will be no error in the method. Therefore, by linearity, we can use $$\big (\mathbf u,\theta ,\phi -\varPi _{Y_{h}}\phi ,\lambda -\varPi _{X_{h}}\lambda \big )$$ as the exact solution in Corollary 5.3 and the numerical solution will be $$\big (\mathbf u^{h},\theta ^{h},\phi ^{h}-\varPi _{Y_{h}}\phi ,\lambda ^{h}-\varPi _{X_{h}}\lambda \big )$$. Consequently, we can substitute $$\|s\phi \|_{1/2,\varGamma }+\|\lambda \|_{-1/2,\varGamma }$$ by $$\| s(\phi -\varPi _{Y_{h}}\phi )\|_{1/2,\varGamma }+\|\lambda -\varPi _{X_{h}}\lambda \|_{-1/2,\varGamma }$$ in the right-hand side of (5.16a)–(5.16b). If we now apply Proposition 4.3 to Corollary 5.3, we may obtain the following estimates in the time domain. Corollary 5.4 If the exact solution quadruple satisfies   $$ \big(\mathbf U, \varTheta, \mathcal{L}^{-1} \{\phi \}, \mathcal{L}^{-1} \{\lambda \}\big ) \in W_{+}^{4}(\mathbf H^{1}(\varOmega_{-})) \times W_{+}^{4} (H^{1}(\varOmega_{-}) )\times W_{+}^{5} ( H^{1/2} (\varGamma) ) \times W_{+}^{4} ( H^{-1/2} (\varGamma) ) $$then   $$ \mathcal{L}^{-1}\big \{\big(\mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)\big \} \in \mathcal{C}\big(\mathbb{R}; \mathbf H^{1}(\varOmega_{-}) \times H^{1}(\varOmega_{-}) \times H^{1}\big(\mathbb{R}^{3} \setminus \varGamma\big)\big) $$is causal and for $$t \geq 0,$$  \begin{align*} |\!|\!|\mathcal{L}^{-1}\big\{\big( \mathbf e^{h}_{\mathbf u}, e^{h}_{\theta}, {e^{h}_{v}}\big)\big\}|\!|\!|_{1} + \| \mathcal{L}^{-1} \big\{ \phi^{h} - \phi\big\} \|_{ 1/2, \varGamma} \leq \; & C t^{2}\max \big\{1, t^{4} \big\} ~g_{h}(t), \\ \|\mathcal{L}^{-1}\big\{\lambda^{h} - \lambda\big\} \|_{-1/2, \varGamma} \leq\; & C{t^{3/2}} \max \big\{1, t^{7/2} \big\}~g_{h}(t), \end{align*}where   \begin{align*} g_{h}(t) :=\; & {\int_{0}^{t}} \!\big ( \| \mathcal{P}_{4} \big( \mathcal{L}^{-1} \big\{ \dot{\phi}- \varPi_{Y_{h}} \dot{\phi}\big\}\big) (\tau) \|_{1/2, \varGamma} + \| \mathcal{P}_{4} \big( \mathcal{L}^{-1} \big\{ \lambda -\varPi_{X_{h}} \lambda \big\}\big) (\tau) \|_{-1/2, \varGamma} \big )\, \textrm{d} \tau\\ &+ {\int_{0}^{t}} \! \big ( \|\mathcal{P}_{4}\big( \ddot{ \mathbf U} - \mathbf P_{h}\ddot{\mathbf U}\big) (\tau) \|_{1, \varOmega_{-}} + \|\mathcal{P}_{4} \big( \dot{\mathbf U} - \mathbf P_{h} \dot{\mathbf U}\big) (\tau) \|_{1, \varOmega_{-}} \big )\, \textrm{d} \tau \\ &+ {\int_{0}^{t}} \! \big(\|\mathcal{P}_{4} ( \mathbf U - \mathbf P_{h}\mathbf U ) (\tau) \|_{1, \varOmega_{-}} + \| \mathcal{P}_{4} ( \varTheta - Q_{h}\varTheta ) (\tau) \|_{1, \varOmega_{-}} \big )\, \textrm{d} \tau . \end{align*} 6. Computational aspects 6.1 Convolution Quadrature We present a very brief description of the procedure used to obtain a full discretization using multistep-based Convolution Quadrature (CQ). The process was devised by Lubich (1988a,b), and was employed originally for treating convolutional boundary integral equations (Lubich & Schneider, 1992; Lubich, 1994). In recent times it has become a very powerful tool for the discretization of time-domain problems. The present description is by no means comprehensive and is provided only for the sake of completeness; the interested reader is referred to Hassell & Sayas (2016a), where very detailed descriptions of the theory and implementation for both multistep and multistage flavors of CQ are given. Suppose that   $$ \dim \,\mathbf{V}_{h} = N_{1}, \quad \dim \,W_{h} = N_{2}, \quad \dim\, Y_{h} = M_{1} \quad \textrm{and}\quad \dim \,X_{h}= M_{2}, $$and let   $$ \big\{\boldsymbol{\mu}_{j}\big\}_{j = 1}^{N_{1}},\quad \big\{ \vartheta_{j}\big\}_{j=1}^{N_{2}},\quad \big\{ \varphi_{j} \big\}_{j=1}^{M_{1}} \quad \textrm{and }\quad \big\{ \eta_{j}\big\}_{j=1}^{M_{2}} $$be the basis functions of the spaces $$\mathbf{V}_{h},\; W_{h},\; Y_{h} $$ and $$X_{h} $$, respectively. We choose a time step $$\varDelta t> 0, $$ and let us consider the uniform grid in time $$t_{n}:= n \varDelta t$$ for $$n\geq 0.$$ We define $$\mathbf A(s) \in \mathbb{C}^{(N_{1}+N_{2}+M_{1}+M_{2})\times (N_{1}+N_{2}+M_{1}+M_{2})}$$ to be the stiffness matrix of equation , where A(s) is a matrix-valued function of $$ s \in \mathbb{C}_{+}$$ whose structure is depicted in Fig. 1. Fig. 1. View largeDownload slide The linear system associated to the discretization has the block structure of the schematic. The elastic and thermal unknowns and the acoustic unknowns are weakly coupled, reflecting the physical fact that the systems communicate only through boundary interactions between the acoustic and elastic variables. Fig. 1. View largeDownload slide The linear system associated to the discretization has the block structure of the schematic. The elastic and thermal unknowns and the acoustic unknowns are weakly coupled, reflecting the physical fact that the systems communicate only through boundary interactions between the acoustic and elastic variables. The data are sampled in time and tested to define vectors $$\mathbf f_{n} \in \mathbb{R}^{(N_{1}+N_{2}+M_{1}+M_{2})}$$:   $$\begin{aligned} f_{n, i} :=\;& (D_{1}(t_{n}), \boldsymbol{\mu}_{i} )_{\varOmega_{-}}, \quad & i =& \ 1, \ldots, N_{1},\\ f_{n, i} :=\;& (D_{2}(t_{n}), \vartheta_{i})_{\varOmega_{-}}, \quad & i =& \ N_{1}+1, \ldots, N_{1}+N_{2},\\ f_{n, i} :=\;& \langle D_{3}(t_{n}), \varphi_{i}\rangle_{\varGamma}, \quad & i =& \ N_{1}+N_{2} +1, \ldots, N_{1}+N_{2} +M_{1},\\ f_{n, i} :=\; & \langle D_{4}(t_{n}), \eta_{i}, \rangle_{\varGamma}, \quad & i =& \ N_{1}+N_{2}+M_{1}+1, \ldots, N_{1}+N_{2}+M_{1}+M_{2}, \end{aligned}$$ where $$ D_{i}(t) = \mathcal{L}\{ d_{i} \}, i =1, \ldots , 4 $$ in Theorem 4.4. The CQ discretization of (5.1) starts with the Taylor expansion   \begin{align} \mathbf{A}\left( \frac{ \gamma(z)}{\varDelta t}\right) = \sum_{n=0}^{\infty} \mathbf A_{n} (\varDelta t) z^{n}, \quad \gamma (z) = \frac{\alpha_{0} + \cdots +\alpha_{k} z^{-k}}{\beta_{0} + \cdots +\beta_{k} z^{-k}}, \end{align} (6.1)where $$\gamma (z)$$ characterizes the underlying k-multistep method, and is, therefore, usually referred to as the characteristic function of the linear multistep method. For the discretization of (5.1), we seek the sequence of vectors $$\mathbf{b}_{n} \in \mathbb{R}^{(N_{1}+N_{2}+M_{1}+M_{2})}$$ given by the recurrence   \begin{align} \mathbf A_{0} (\varDelta t) ~\mathbf b_{n} = ~\mathbf f_{n} - \sum_{m=1}^{n} \mathbf A_{m} (\varDelta t)~ \mathbf b_{n-m}, \quad n \geq 0. \end{align} (6.2)The coefficients $$A_{n}(\varDelta t)$$ can be computed by means of Cauchy’s integral formula   $$ \mathbf A_{m}(\varDelta t) = \frac{1}{m!}\frac{\textrm{d}^{(m)}}{\textrm{d}z^{(m)}}\left(\mathbf A(\gamma(z)/\varDelta t)\right)|_{z=0} = \frac{1}{2\pi i}\oint_{C} \zeta^{-m-1}\mathbf A(\gamma(\zeta)/\varDelta t)\,\textrm{d}\zeta. $$For implementation purposes, the integration contour C is taken to be a circle with radius $$R_{C}$$ dependent on the number of terms in the expansion and the specific value of the computer’s machine epsilon (Hassell & Sayas, 2016a). This choice of contour allows for fast and accurate computation of the coefficients exploiting the properties of the trapezoidal rule and the fast Fourier transform. If the solution of (6.2) assumes the form $${\mathbf{{b}}}_{n} = \big (b_{n,1}, \ldots , b_{n, (N_{1}+N_{2}+M_{1}+M_{2})}\big )$$, then the Galerkin solutions of (5.1) at $$t_{n}$$ are given by   $$\begin{aligned} \mathbf{u^{h}_{n}} = \;& \sum_{j=1}^{N_{1}} b_{n,j} \boldsymbol{\mu}_{j}, & \qquad{\theta_{n}^{h}} =\;& \sum_{j=N_{1}+1}^{N_{1}+N_{2}} b_{n,j} \vartheta_{j},\\[-8pt] \nonumber\\{\phi^{h}_{n}} =\;& \sum_{j=N_{1}+N_{2}+1}^{N_{1}+N_{2}+M_{1}} b_{n,j} \varphi_{j}, &\qquad{\lambda^{h}_{n}} =\;& \sum_{j=N_{1}+N_{2}+M_{1}+1}^{N_{1}+N_{2}+M_{1}+M_{2}} b_{n, j} \eta_{j}. \end{aligned}$$ 6.2 A combined approach for time evolution The linear system arising from the discretization and depicted in Fig. 1 can be thought of as having the block structure   $$ \left[\begin{array}{cc} {\mathbf{FEM}}(s) & s\rho_{f}(\textrm N\varGamma)_{h}^\textrm{T} \\ -s\rho_{f}(\textrm N\varGamma)_{h} &{\mathbf{BEM}}(s) \end{array}\right] \left[\begin{array}{c} \left[\begin{array}{c}\mathbf u^{h} \\ \theta^{h} \end{array}\right] \\[2.5ex] \left[\begin{array}{c} \lambda^{h} \\ \phi^{h} \end{array}\right] \end{array}\right] = \left[\begin{array}{c} \left[\begin{array}{c} \!\!\!-s\rho_{f}{\varGamma_{h}^\textrm{T}}\beta^{h} \!\!\!\\ \eta^{h} \end{array}\right] \\[2.5ex] \left[\begin{array}{c} 0 \\ \rho_{f}\alpha^{h} \end{array}\right] \end{array}\right], $$where the sparse finite element block   \begin{align*} \mathbf{FEM}(s):=\;&\; s^{2}\left[\begin{array}{cc}(\rho_{\varSigma} \mathbf u_{j},\boldsymbol v_{i})_{\varOmega_{-}} & 0 \\ 0 & 0 \end{array}\right] + s\left[\begin{array}{cc} 0 & 0 \\ -(\boldsymbol\eta\mathbf u_{j}, \nabla v_{i})_{\varOmega_{-}} & (\theta_{j},v_{i})_{\varOmega_{-}}\end{array}\right] \\ &+\left[\begin{array}{cc} (\mathbf C\boldsymbol\varepsilon(\mathbf u_{j}),\boldsymbol\varepsilon(\boldsymbol v_{i}))_{\varOmega_{-}} & -(\boldsymbol\zeta\theta_{j},\boldsymbol\varepsilon(\boldsymbol v_{i}))_{\varOmega_{-}} \\ 0 & (\boldsymbol\kappa\nabla\theta_{j},\nabla v_{i})_{\varOmega_{-}}\end{array}\right] \end{align*}contains mass and stiffness matrices as well as first-order terms related to the elastic and thermal unknowns. The boundary element block BEM(s) contains the Galerkin discretization of the operators of the acoustic Calderón calculus and the coupling trace matrix $$(\textrm N\varGamma )_{h}$$ is the discretization of the bilinear form arising from the duality pairing $$\big \langle \mathbf u^{h}\cdot \boldsymbol \nu ,\chi ^{h}\big \rangle _{\varGamma }$$. Even if a CQ approach can be applied to the entire system using the expansion (6.1) on the global matrix and solving the resulting linear system (5.13), it is common practice to decouple the computations of the finite element and boundary element unknowns via a Schur complement strategy. The decoupled boundary element unknowns are then evolved in time using CQ (via (6.1) and (5.13)), while the same underlying multistep scheme is used for the finite element unknowns. This process was first described in the study by Banjai & Sauter (2009) and is explained in detail in the study by Hassell & Sayas (2016b), where it is used in the context of purely acoustic waves. 6.3 Numerical experiments In order to test numerically the formulations of the previous sections, computational convergence studies were performed in both frequency and time domains for two-dimensional test problems. Moreover, we explore numerically the case when the Lamé parameters or the thermal diffusivity and expansion are nonconstant tensors. We emphasize that the goal of the computations presented here is mainly to provide a proof of concept of the suggested discretization and to highlight the fact that the discretization can be readily implemented with only minor additions to existing code. The previous analysis remains valid in three dimensions and the implementation of the discretization in that case can be done following completely analogous steps. The computational domain. For the convergence studies, the interior domain $$\varOmega _{-}$$ where the thermoelastic equations were imposed was the polygon depicted in Fig. 2. The domain was generated and meshed using MATLAB’s pdetool. All the mesh refinements were done using the refinement capabilities of the partial differential equation toolbox. Fig. 2. View largeDownload slide Interior geometry used in the numerical experiments for both frequency-domain and time-domain studies. The domain was generated and meshed using MATLAB’s pdetool and refined uniformly using pdetool’s refinement capabilities. Fig. 2. View largeDownload slide Interior geometry used in the numerical experiments for both frequency-domain and time-domain studies. The domain was generated and meshed using MATLAB’s pdetool and refined uniformly using pdetool’s refinement capabilities. Table 1 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates (e.c.r) in the frequency domain for k = 1. The maximum length of the panels used to discretize the boundary is denoted by h k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  View Large Table 1 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates (e.c.r) in the frequency domain for k = 1. The maximum length of the panels used to discretize the boundary is denoted by h k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  5.016 e−2  7.292 e−3  1.293  1.675 e−2  1.255  6.397 e−4  1.656  8.733 e−2  1.203  3.746 e−2  0.988  2.508 e−2  2.272 e−3  1.683  5.344 e−3  1.648  1.837 e−4  1.799  3.297 e−2  1.405  1.876 e−2  0.976  1.254 e−2  6.099 e−4  1.897  1.447 e−3  1.885  4.824 e−5  1.929  1.314 e−2  1.327  9.383 e−3  0.996  6.27 e−3  1.556 e−4  1.971  3.703 e−4  1.966  1.223 e−4  1.980  5.961 e−3  1.141  4.692 e−3  1.000  View Large Approximation errors. As a measurement of the accuracy of the approximations, the difference between the manufactured solution and the approximate finite element solution was measured in the $$L^{2}(\varOmega _{-})$$- and $$H^{1}(\varOmega _{-})$$-norms for the elastic and thermal variables $$\mathbf u^{h}$$ and $$\theta ^{h}$$. For the acoustic unknown $$v^{h}$$, the approximate solution was sampled at 25 randomly placed points in $$\varOmega _{+}$$ and the maximum absolute difference from the exact solution was taken as a measure of the error. In the time-domain experiments these measurements were done for a final time t = 1.5. Table 2 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 2. The maximum length of the panels used to discretize the boundary is denoted by h k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  Table 2 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 2. The maximum length of the panels used to discretize the boundary is denoted by h k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  5.016 e−2  6.676 e−6  3.570  1.181 e−5  3.442  1.214 e−5  3.004  8.708 e−4  2.013  1.630 e−3  1.983  2.508 e−2  5.590 e−7  3.578  1.207 e−6  3.290  1.517 e−6  3.000  2.172 e−4  2.003  4.093 e−4  1.993  1.254 e−2  4.630 e−8  3.594  1.331 e−7  3.181  1.897 e−7  2.999  5.426 e−5  2.001  5.426 e−5  1.997  6.27 e−3  3.793 e−9  3.609  1.550 e−8  3.103  2.373 e−8  2.999  1.356 e−8  2.001  2.566 e−5  1.999  Table 3 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 3. The maximum length of the panels used to discretize the boundary is denoted by h k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  Table 3 The experiments were run using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. This table shows the relative errors and estimated convergence rates in the frequency domain for k = 3. The maximum length of the panels used to discretize the boundary is denoted by h k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  h  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  1 e−1  6.847 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  5.016 e−2  3.869 e−8  4.145  9.804 e−8  4.138  2.886 e−7  3.983  7.701 e−7  3.631  5.044 e−5  2.994  2.508 e−2  2.279 e−9  4.085  5.794 e−9  4.081  1.810 e−8  3.995  7.600 e−8  3.341  6.312 e−6  2.998  1.254 e−2  1.375 e−10  4.051  3.502 e−10  4.048  1.132 e−9  3.999  8.504 e−9  3.160  7.892 e−7  3.000  6.27 e−3  8.468 e−12  4.021  2.141 e−11  4.032  7.076 e−11  4.000  1.011 e−9  3.072  9.866 e−8  3.000  Table 4 Frequency-domain p-convergence studies. The experiments were run on a fixed mesh with parameter h = 0.1 using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements with k = 1, … , 5. The relative error is shown as a function of the number of degrees of freedom (Ndof) h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  Table 4 Frequency-domain p-convergence studies. The experiments were run on a fixed mesh with parameter h = 0.1 using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements with k = 1, … , 5. The relative error is shown as a function of the number of degrees of freedom (Ndof) h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  h = 0.1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  Ndof (degree)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  108 (1)  1.787 e−2  —  3.999 e−2  —  2.015 e−3  —  2.011 e−1  —  7.430 e−2  —  394 (2)  7.926 e−5  —  1.284 e−4  —  9.742 e−5  —  3.514 e−3  —  6.446 e−3  —  859 (3)  6.848 e−7  —  1.726 e−6  —  4.564 e−6  —  9.540 e−6  —  4.018 e−4  —  1503 (4)  5.185 e−9  —  1.154 e−8  —  1.503 e−7  —  2.861 e−7  —  2.042 e−5  —  2326 (5)  1.241 e−10  —  3.533 e−10  —  5.814 e−9  —  9.008 e−9  —  8.133 e−7  —  Fig. 3. View largeDownload slide Convergence studies in the frequency domain. Top row and bottom left: Successive mesh refinements were carried out for basis functions with polynomial degrees k = 1, 2 and 3. Bottom right: Approximation errors as a function of the degrees of freedom for basis functions of increasing order over a fixed mesh with parameter h = 0.1. For the color code we refer the reader to the electronic version of the manuscript. Fig. 3. View largeDownload slide Convergence studies in the frequency domain. Top row and bottom left: Successive mesh refinements were carried out for basis functions with polynomial degrees k = 1, 2 and 3. Bottom right: Approximation errors as a function of the degrees of freedom for basis functions of increasing order over a fixed mesh with parameter h = 0.1. For the color code we refer the reader to the electronic version of the manuscript. Physical parameters. The following values of the physical parameters are functions only of space and were used equally for both series of experiments. They are chosen for validation and expository purposes only and do not correspond to any relevant physical material. For the entries of the tensors we make use of the symmetries and of Voigt’s notation (Gurtin, 1973) to shorten the subscripts. 1. Density of the elastic solid and Lamé parameters:   \begin{align} \rho_{\varSigma} = 5 + \sin{(x)}\sin{(y)},\qquad \lambda = 2, \qquad \mu =3. \end{align} (6.3) 2. Table 5 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Table 5 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  BDF2. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  4.486 e−3  —  1.217 e−2  —  5.381 e−3  —  2.975 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  1.130 e−3  1.989  3.337 e−3  1.867  1.358 e−3  1.987  1.434 e−1  1.053  1.121 e−1  0.986  4.697 e−3 (1.254 e−2)  7.293 e−5  1.990  2.214 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  1.827 e−5  1.997  5.569 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Table 6 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  Table 6 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  BDF2. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.624 e−3  —  7.798 e−4  —  2.917 e−4  —  1.297 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  6.471 e−4  2.486  2.004 e−4  1.960  3.640 e−5  3.002  3.370 e−3  1.944  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  1.571 e−4  2.043  5.062 e−5  1.985  4.550 e−6  3.000  8.524 e−4  1.983  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  3.891 e−5  2.013  1.271 e−5  1.994  5.692 e−7  2.999  2.137 e−4  1.996  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  9.701 e−6  2.004  3.182 e−6  1.998  7.119 e−8  2.999  5.347 e−5  1.999  7.697 e−5  1.999  Table 7 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  Table 7 Time-domain convergence results for BDF2-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  BDF2. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  3.631 e−3  —  7.616 e−4  —  1.368 e−5  —  7.737 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  6.480 e−4  2.486  1.995 e−4  1.933  8.649 e−7  3.983  2.140 e−3  1.854  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  1.571 e−4  2.044  5.059 e−5  1.980  5.423 e−8  3.995  5.506 e−4  1.959  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  3.892 e−5  2.013  1.270 e−5  1.993  3.392 e−9  3.999  1.386 e−4  1.990  2.368 e−6  3.000  Table 8 Time-domain convergence results for BDF2-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  Table 8 Time-domain convergence results for BDF2-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  BDF2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  7.793 e−3  —  1.231 e−2  —  5.184 e−3  —  2.975 e−1  —  2.222 e−1  —  1.875 e−2 (394)  2.775 e−3  1.489  7.725 e−4  3.994  3.275 e−4  3.984  1.258 e−2  4.563  1.940 e−2  3.518  9.375 e−3 (859)  7.955 e−4  1.803  1.980 e−4  1.964  4.061 e−5  3.012  1.916 e−3  2.715  1.265 e−3  3.938  4.687 e−3 (1503)  2.072 e−4  1.941  5.035 e−5  1.975  9.408 e−6  2.110  4.905 e−4  1.966  1.125 e−4  3.489  2.344 e−3 (2326)  5.258 e−5  1.978  1.267 e−5  1.991  2.329 e−6  2.014  1.236 e−4  1.988  2.355 e−5  2.259  1.172 e−3 (3328)  1.323 e−5  1.991  3.175 e−6  1.996  5.795 e−7  2.007  3.100 e−5  1.995  5.825 e−6  2.015  Thermal expansion tensor $$\boldsymbol \zeta $$:   \begin{align} \boldsymbol\zeta_{1}\leftrightarrow\, \boldsymbol\zeta_{11} \!=\! \sin{(x)}+\cos{(y)}, \quad \boldsymbol\zeta_{2} \leftrightarrow \,\boldsymbol\zeta_{22} \!=\! -\sin{(y)}, \quad \boldsymbol\zeta_{3} \leftrightarrow\, \boldsymbol\zeta_{12} \!=\! \boldsymbol\zeta_{21} \!=\! \cos{(x)}. \end{align} (6.4) 3. Thermal diffusivity tensor $$\boldsymbol \kappa $$:   \begin{align} \boldsymbol\kappa_{1}\leftrightarrow\, \boldsymbol\kappa_{11} = 10 + x^{2}, \quad \boldsymbol\kappa_{2} \leftrightarrow \,\boldsymbol\kappa_{22} = 10+ y, \quad \boldsymbol\kappa_{3} \leftrightarrow\, \boldsymbol\kappa_{12}= \boldsymbol\kappa_{21} = 0. \end{align} (6.5) 4. The components of the tensor $$\boldsymbol \eta $$ were chosen to be   \begin{align} \boldsymbol\eta_{1}\leftrightarrow\, \boldsymbol\eta_{11} = 1, \quad \boldsymbol\eta_{2} \leftrightarrow \,\boldsymbol\eta_{22} = x + y, \quad \boldsymbol\eta_{3} \leftrightarrow\, \boldsymbol\eta_{12}= \boldsymbol\eta_{21} = 5+x+y. \end{align} (6.6) Convergence studies in the frequency domain. We first verify the results in the frequency domain. We proceed by the method of manufactured solutions using the functions  \begin{align*} \mathbf u :=&\, \big(x^{3}+xy+y^{3},\sin{(x)}\cos{(y)}\big), \qquad \theta := \sin^{2}{(\pi x)}\sin^{2}{(y)}, \\ v :=&\, \tfrac{i}{4}H_{0}^{(1)}(isr), \qquad \qquad\qquad\qquad\quad\;\;\, r = \sqrt{x^{2}+y^{2}}, \end{align*}together with the parameters defined in (6.3)–(6.6). Right-hand-side load vectors and boundary conditions were constructed accordingly. For the numerical experiments, Lagrangian $$\mathcal P_{k}$$ finite elements were used for the elastic and thermal unknowns, while Galerkin $$\mathcal P_{k}/\mathcal P_{k-1}$$ continuous/discontinuous boundary elements were used for the acoustic potential v. Convergence studies for spatial refinements with a fixed polynomial degree (h-convergence) and increasing degree of polynomial approximation with a fixed mesh size (p-convergence) were performed for s = 2.8i. The results of the mesh-refinement experiments are shown in Tables 1, 2 and 3. Table 4 contains the results for a fixed mesh with increasing polynomial degree for the basis functions. The convergence plots for all the simulations are displayed in Fig. 3. Convergence studies in the time domain. In a way analogous to the previous section, the numerical experiments were carried out using the physical parameters and coefficients given in (6.3)–(6.6) and with manufactured solutions using the functions  \begin{align*} \mathbf u :=&\, \textrm T(t) \big(x^{3}+xy+y^{3},\sin{(x)}\cos{(y)}\big), \qquad \theta := \textrm T(t) \sin^{2}{(\pi x)}\sin^{2}{(y)}, \\ v :=&\,\mathcal{L}^{-1}\left\{iH^{(1)}_{0}(i s r)\,\mathcal{L}\{\mathcal{H}(t)\sin(3t)\} \right\}, \qquad r := \sqrt{x^{2}+y^{2}}, \end{align*} Fig. 4. View largeDownload slide Time-domain convergence studies for the BDF2-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Fig. 4. View largeDownload slide Time-domain convergence studies for the BDF2-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Table 9 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  View Large Table 9 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 1 Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  Trapezoidal rule. k = 1  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.497 e−3  —  1.222 e−2  —  5.381 e−3  —  2.977 e−1  —  2.221 e−1  —  1.875 e−2 (5.016 e−2)  1.648 e−3  1.979  3.347 e−3  1.868  1.358 e−3  1.987  1.434 e−1  1.054  1.121 e−1  0.986  9.375 e−3 (2.508 e−2)  4.145 e−4  1.991  8.731 e−4  1.939  3.407 e−4  1.995  6.996 e−2  1.036  5.624 e−2  0.996  4.697 e−3 (1.254 e−2)  1.038 e−4  1.997  2.220 e−4  1.975  8.528 e−5  1.998  3.458 e−2  1.017  2.814 e−2  0.999  2.344 e−3 (6.270 e−3)  2.596 e−5  1.999  5.585 e−5  1.991  2.133 e−5  1.999  1.721 e−2  1.006  1.407 e−2  1.000  View Large Table 10 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  View Large Table 10 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 2 Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  Trapezoidal rule. k = 2  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.150 e−4  —  2.709 e−4  —  2.917 e−4  —  1.069 e−2  —  1.934 e−2  —  1.875 e−2 (5.016 e−2)  1.594 e−4  1.948  5.531 e−5  2.292  3.640 e−5  3.002  2.668 e−3  2.002  4.890 e−3  1.983  9.375 e−3 (2.508 e−2)  4.114 e−5  1.954  1.301 e−5  2.088  4.550 e−6  3.000  6.662 e−4  2.002  1.228 e−3  1.993  4.697 e−3 (1.254 e−2)  1.036 e−5  1.989  3.202 e−6  2.022  5.692 e−7  2.999  1.664 e−4  2.001  3.076 e−4  1.997  2.344 e−3 (6.270 e−3)  2.596 e−6  1.997  7.974 e−7  2.005  7.119 e−8  2.999  4.159 e−5  2.001  7.697 e−5  1.999  View Large Table 11 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  View Large Table 11 Time-domain convergence results for trapezoidal-rule-based CQ with combined h and $$\varDelta t$$ refinements. In every successive refinement level the sizes of the time step and the mesh parameter were halved. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 and polynomial degree k = 3 Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  Trapezoidal rule. k = 3  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (h)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (1.000 e−1)  6.108 e−4  —  2.027 e−4  —  1.368 e−5  —  2.204 e−3  —  1.205 e−3  —  1.875 e−2 (5.016 e−2)  1.601 e−4  1.932  5.090 e−5  1.994  8.650 e−7  3.983  5.550 e−4  1.990  1.513 e−4  2.994  9.375 e−3 (2.508 e−2)  4.122 e−5  1.958  1.274 e−5  1.998  5.424 e−8  3.995  1.390 e−4  1.998  1.894 e−5  2.998  4.697 e−3 (1.254 e−2)  1.037 e−5  1.991  3.186 e−6  2.000  3.392 e−9  3.999  3.475 e−5  1.999  2.368 e−6  3.000  View Large Table 12 Time-domain convergence results for trapezoidal-rule-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  View Large Table 12 Time-domain convergence results for trapezoidal-rule-based CQ. The experiments were run with a fixed mesh using $$\mathcal P_{k}$$ Lagrangian finite elements and $$\mathcal P_{k}/\mathcal P_{k-1}$$ boundary elements. In every successive refinement level the size of the time step was halved and the polynomial degree of the space refinement increased by 1. The table shows the relative errors and estimated convergence rates measured for a final time t = 1.5 as a function of the time step $$\varDelta t$$ and the Ndof used in the spatial discretization Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  Trapezoidal rule  $$L^{2}(\varOmega _{-})$$  $$H^{1}(\varOmega _{-})$$  $$\varDelta t$$ (Ndof)  $${E^{v}_{h}}$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h,\kappa }$$  e.c.r.  $$E^{\mathbf u}_{h}$$  e.c.r.  $$E^{\theta }_{h}$$  e.c.r.  3.75 e−2 (108)  5.620 e−3  —  1.218 e−2  —  5.213 e−3  —  2.976 e−1  —  2.221 e−1  —  1.875 e−2 (394)  8.283 e−4  2.762  2.713 e−4  5.489  2.934 e−4  4.151  1.064 e−2  4.805  1.934 e−2  3.522  9.375 e−3 (859)  2.107 e−4  1.975  5.085 e−5  2.416  1.660 e−5  4.144  4.958 e−4  4.424  1.209 e−3  4.000  4.687 e−3 (1503)  5.278 e−5  1.997  1.272 e−5  1.999  2.349 e−6  2.821  1.242 e−4  1.997  6.549 e−5  4.206  2.344 e−3 (2326)  1.320 e−5  1.996  3.184 e−6  1.999  5.770 e−7  2.026  3.107 e−5  1.999  6.286 e−6  3.381  1.172 e−3 (3328)  3.300 e−6  2.000  7.956 e−7  2.000  1.442 e−7  2.001  7.770 e−6  2.000  1.451 e−6  2.115  View Large Fig. 5. View largeDownload slide Time-domain convergence studies for the trapezoidal rule-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. Fig. 5. View largeDownload slide Time-domain convergence studies for the trapezoidal rule-based time stepping scheme. For the color code we refer the reader to the electronic version of the manuscript. where $$\mathcal L\{\cdot \}$$ is the Laplace transform, the time factor T(t) is given by   \begin{align} \textrm T := \mathcal H(t)\big(t^{2}+2t\big), \end{align} (6.8a)and $$\mathcal H(t)$$ is the $$\mathcal C^{5}$$ approximation to Heaviside’s step function   $$ \mathcal H(t)\!:= \!t^{5}(1-5(t-1)\!+\!15(t-1)^{2}\!-\!35(t-1)^{3}\!+\!70(t-1)^{4}\!-\!126(t-1)^{5})\chi_{[0,1]}(t)\!+\!\chi_{[1,\infty)}(t). $$Two kinds of experiments were carried out using the same geometry as in the frequency domain. First, for a spatial discretization with fixed polynomial degree, successive dyadic refinements in both mesh size h and time step $$\varDelta t$$ were carried out (h-refinement). The experiment was repeated for polynomial degrees k = 1, 2 and 3 starting with a spatial mesh with parameter $$h=1 \times 10^{-1}$$ and time step $$\varDelta t = 3.75 \times 10^{-2}$$. The second experiment corresponds to p-refinements in space and consisted in using a fixed spatial mesh, starting with a polynomial discretization of degree k = 1 in space and a time step $$\varDelta t = 3.75 \times 10^{-2}$$. With every successive dyadic refinement of $$\varDelta t$$, the degree of the polynomial interpolant was increased by 1. The initial mesh size of $$h=5.016 \times 10^{-2}$$ corresponds to the second level of refinement used for the h-refinement experiments. These space-time refinement strategies highlight the global order of convergence of the method, which is expected to be asymptotically limited by the order of the multistep scheme used for time discretization. Both strategies were tried for BDF2 and trapezoidal-rule-based time discretizations. The results for the studies using BDF2 are shown in Tables 5, 6, 7 (h-refinement) and 8 (p-refinement). These results are summarized in the convergence plot of Fig. 4. Similarly, the results for the experiments using trapezoidal-rule time stepping are shown in Tables 9, 10, 11 (h-refinement) and 12 (p-refinement), all condensed on the convergence plots shown in Fig. 5. Table 13 Number of finite element degrees of freedom required to represent a single scalar function defined on the mesh depicted in Fig. 2. The number increases depending on the chosen refinement strategy, with the slowest growth rate being the one associated to p-refinements Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Table 13 Number of finite element degrees of freedom required to represent a single scalar function defined on the mesh depicted in Fig. 2. The number increases depending on the chosen refinement strategy, with the slowest growth rate being the one associated to p-refinements Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Growth in FEM DOF  Refinement level  Refinement strategy  1  2  3  4  5  6  h-refinement, k = 1  108  394  1503  5869  23193  92209  h-refinement, k = 2  394  1503  5869  23193  92209  161225  h-refinement, k = 3  859  3328  13099  51973  207049  413537  p-refinement  108  394  859  1503  2326  3328  Fig. 6. View largeDownload slide Snapshots at times t = 0.6, 1.3, 2 of the interaction between an acoustic wave and a thermoelastic obstacle. The acoustic field is depicted in the top row, the norm of the elastic displacement in the middle row (black denotes no displacement) and the temperature variations in the bottom row (black represents zero variation, shades of red and blue represent positive and negative variations, respectively). The reader is referred to the on-line version of the paper for the color code. Fig. 6. View largeDownload slide Snapshots at times t = 0.6, 1.3, 2 of the interaction between an acoustic wave and a thermoelastic obstacle. The acoustic field is depicted in the top row, the norm of the elastic displacement in the middle row (black denotes no displacement) and the temperature variations in the bottom row (black represents zero variation, shades of red and blue represent positive and negative variations, respectively). The reader is referred to the on-line version of the paper for the color code. Depending on the refinement strategy, the number of degrees of freedom (DOF) required to approximate the system increases quickly, especially for h-refinements with a higher-order polynomial basis. Table 13 shows the number of unknowns associated to a single scalar FEM function represented in the grid shown in Fig. 2. The increase in computational requirements imposed by h-refinement makes some asymptotic properties of the scheme difficult to observe following this strategy. In particular, the smoothing properties of the parabolic part of the system introduce superconvergent behavior on the thermal unknowns during the pre-asymptotic regime. As can be seen in the p-refinement experiments (cf. Figs 4 and 5, bottom right) the convergence stabilizes to the predicted rate for relatively small time steps, after 5 refinement levels. The number of spatial degrees of freedom required to achieve such a discretization level by h-refinements causes the true convergence rate to be observable using only a p-refinement strategy. One example. We conclude with a simple illustrative example in two dimensions showing the interaction between the plane wave   $$ v^{\textrm{inc}}= 3\chi_{[0,0.3]}(88\tau)\sin{(88\tau)},\quad \tau:= t-\mathbf r\cdot\mathbf d,\quad \mathbf r:=(x,y),\quad \mathbf d := (1,5)/\sqrt{26}, $$and a pentagonal scatterer with mass density given by   $$ \rho_{\varSigma} = 15 + 40e^{-49\, r^{2}}, \qquad r := \sqrt{x^{2}+y^{2}}. $$ The values of the elastic parameters, thermal diffusivity $$\boldsymbol \kappa $$, thermoelastic expansion tensors $$\boldsymbol \zeta $$ and $$\boldsymbol \eta $$ were the same as those used for the convergence experiments in the previous paragraphs and given in equations (6.3)–(6.6). The simulation used $$\mathcal P_{2}$$ Lagrangian finite elements on a grid with mesh parameter $$h=7 \times 10^{-3}$$ and 36096 elements. The inherited boundary element grid had 496 panels and a grid parameter of $$h=9.1 \times 10^{-3}$$, and $$\mathcal P_{2}/\mathcal P_{1}$$ continuous/discontinuous Galerkin boundary elements were used. Trapezoidal-rule-based discretization was applied in time with a time step $$\varDelta t=1\times 10^{-2}$$. Some snapshots of the simulation are shown in Fig. 6. Acknowledgements The authors would like to thank the referees for their detailed comments and suggestions, which greatly improved the quality of this communication. References Bamberger, A. & Ha-Duong, T. ( 1986) Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique (I). Math. Meth. Appl. Sci. , 8, 405– 435. Google Scholar CrossRef Search ADS   Banjai, L., Lubich, C. & Sayas F. J. 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