Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method

TL;DR

This work develops a non-overlapping domain decomposition solver for Helmholtz-type acoustic problems using a Generalized Optimized Schwarz Method (GOSM) tailored to fully discrete FEM-BEM couplings. It establishes well-posed substructured formulations for classical FEM-BEM couplings (Costabel, Johnson-Nédélec, Bielak-MacCamy) and proves that, under a physical dissipation sign condition, the skeleton operator Id+ΠS is contractive, enabling guaranteed geometric convergence of a Richardson iteration even with cross-points. The analysis links kernel dimensions of the initial and substructured problems and provides an explicit scattering operator S and exchange operator Π to couple subdomains efficiently; it also discusses h-robustness via appropriate transmission operators. Numerical experiments confirm h-independence and rapid convergence for nonlocal transmissions (e.g., Schur/composed Yukawa) across homogeneous and heterogeneous media, including configurations with cross-points and weakly imposed boundary conditions, highlighting practical performance for large-scale 3D problems and complex geometries.

Abstract

The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-Nédélec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.

Figures (11)

• Figure 1: Two examples of the layer $\omega_s \subset \omega$ to compute the Schur complement based transmission operator $\mathrm{T}_{S}$ for a subdomain $\omega$. The layer $\omega_s$ is the shaded red region, and its boundary is the union of $\Gamma_{\omega} \coloneq \partial \omega$ (solid blue line) and of $\Gamma_s \coloneqq \partial \omega_s \setminus \Gamma_{\omega}$ (dashed black line)
• Figure 2: Homogeneous test case, which has an explicit expression of the exact solution
• Figure 3: Homogeneous problem with strongly imposed Dirichlet boundary conditions and the Costabel coupling. Left, $N_{it}$ with respect to $N_{\lambda}$ for $\kappa=20$ and $h\kappa = 2\pi/N_{\lambda}$. Right, $N_{it}$ with respect to the wavenumber $\kappa$, for $h^2\kappa^3 = (2\pi/10)^2$
• Figure 4: Homogeneous problem with strongly imposed Neumann boundary conditions and the Costabel coupling. Left, $N_{it}$ with respect to $N_{\lambda}$ for $\kappa=20$ and $h\kappa = 2\pi/N_{\lambda}$. Right, $N_{it}$ with respect to the wavenumber $\kappa$, for $h^2\kappa^3 = (2\pi/10)^2$
• Figure 5: FEM subdomain $\Omega$ with a heterogeneous pocket (left). Amplitude of the total field, for Y-S configuration, Costabel coupling and $k = 10$, $N_{\lambda} = 20$ (right)

Theorems & Definitions (44)

• Example 3.1: Symmetric Costabel coupling Costabel1987SMCMR1052136
• Example 3.2: Johnson-Nédélec coupling JohnsonNedelec1980CBIMR3089444MR2831059
• Example 3.3: Bielak-MacCamy coupling MR700668
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Remark 3.6
• Lemma 5.1
• Lemma 5.2