FEM-BEM mortar coupling for the Helmholtz problem in three dimensions

TL;DR

[The problem] addresses time-harmonic acoustic scattering with variable sound speed in 3D. [The method] introduces a three-field FEM-BEM mortar coupling using an impedance mortar variable $m$ on the interface to couple interior FEM and exterior BEM, yielding a block-structured system. [The analysis] proves a Garding inequality for the coupling form and establishes (quasi-)optimal convergence via a Schatz argument, supported by duality-based regularity results and a $k$-explicit appendix for analytic boundaries. [The results] are complemented by numerical experiments showing stable performance across a range of wavenumbers, including eigenvalue scenarios, and by implementation details demonstrating practical viability with existing solvers and libraries.

Abstract

We present a FEM-BEM coupling strategy for time-harmonic acoustic scattering in media with variable sound speed. The coupling is realized with the aid of a mortar variable that is an impedance trace on the coupling boundary. The resulting sesquilinear form is shown to satisfy a Garding inequality. Quasi-optimal convergence is shown for sufficiently fine meshes. Numerical examples confirm the theoretical convergence results.

Figures (5)

• Figure 1: The four panels depict the four errors in \ref{['computed:quantities']} for $p=1,2,3$ in \ref{['approximation:spaces']} versus the mesh size $h$. The wavenumber $k=1.5\sqrt 3 \pi$ is neither an interior Dirichlet nor a Neumann eigenvalue. $\mathfrak a$ is constant. The solution is given in \ref{['testcase1:solution']}. Top-left panel:$H^1$ error in $\Omega$. Top-right panel:$L^2$ error in $\Omega$. Bottom-left panel:$L^2$ error on $\Gamma$ of the mortar variable times $h^{\frac{1}{2}}$. Bottom-right panel:$L^2$ error on $\Gamma$ times $h^{-\frac{1}{2}}$.
• Figure 2: The four panels show the four errors in \ref{['computed:quantities']} for $p =1,2,3$ in \ref{['approximation:spaces']} versus the mesh size $h$. The wavenumber $k=3 \sqrt 3 \pi$ is a Dirichlet-Laplace eigenvalue. $\mathfrak a$ is constant. The solution is provided in \ref{['testcase1:solution']}. Top-left panel:$H^1$ error in $\Omega$. Top-right panel:$L^2$ error in $\Omega$. Bottom-left panel:$L^2$ error on $\Gamma$ of the mortar variable times $h^{\frac{1}{2}}$. Bottom-right panel:$L^2$ error on $\Gamma$ times $h^{-\frac{1}{2}}$.
• Figure 3: The four panels show the four errors in \ref{['computed:quantities']} for $p=1,2,3$ in \ref{['approximation:spaces']} versus the mesh size $h$. The wavenumber $k=6 \sqrt 3 \pi$ is a Dirichlet-Laplace eigenvalue. $\mathfrak a$ is constant. The solution is provided in \ref{['testcase1:solution']}. Top-left panel:$H^1$ error in $\Omega$. Top-right panel:$L^2$ error in $\Omega$. Bottom-left panel:$L^2$ error on $\Gamma$ of the mortar variable times $h^{\frac{1}{2}}$. Bottom-right panel:$L^2$ error on $\Gamma$ times $h^{-\frac{1}{2}}$.
• Figure 4: The four panels show the four errors in \ref{['computed:quantities']} for different meshes versus the polynomial degree $p$. The wavenumber $k n=3 \sqrt 3 \pi$ is a Dirichlet-Laplace eigenvalue. $\mathfrak a$ is constant. The solution is provided in \ref{['testcase1:solution']}. Top-left panel:$H^1$ error in $\Omega$. Top-right panel:$L^2$ error in $\Omega$. Bottom-left panel:$L^2$ error on $\Gamma$ of the mortar variable times $h^{\frac{1}{2}}$. Bottom-right panel:$L^2$ error on $\Gamma$ times $h^{-\frac{1}{2}}$.
• Figure 5: We depict the four errors in \ref{['computed:quantities']} in the four panel, for different choices of the mesh, versus the polynomial degree. The wavenumber $k n=3 \sqrt 3 \pi$ is a Dirichlet-Laplace eigenvalue. The diffusion parameter $\mathfrak a$ is piecewise constant. The solution is provided in \ref{['testcase2:solution']}. Top-left panel:$H^1$ error in $\Omega$. Top-right panel:$L^2$ error in $\Omega$. Bottom-left panel:$L^2$ error on $\Gamma$ of the mortar variable times $h^{\frac{1}{2}}$. Bottom-right panel:$L^2$ error on $\Gamma$ times $h^{-\frac{1}{2}}$.

Theorems & Definitions (36)

• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Remark 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Remark 3.4
• Theorem 3.5