Optimal convergence rates of an adaptive hybrid FEM-BEM method for full-space linear transmission problems

TL;DR

This work analyzes a hybrid FEM-BEM discretization for full-space linear elliptic transmission problems, achieving a simplification to two sparse FEM solves plus a boundary-integral interpolation. It proves a Céa-type a priori bound and constructs a reliable and efficient a posteriori estimator, then drives adaptivity with a Dörfler-marked loop and newest-vertex refinement. The main theoretical contribution is the proof of optimal convergence rates with respect to the number of elements by verifying the four axioms of adaptivity, complemented by comprehensive 2D numerical experiments on square, L-shaped, and Z-shaped domains. Practically, the framework provides an robust, provably optimal adaptive method for full-space transmission problems leveraging boundary-integral operators and harmonic lifting to couple interior and exterior fields.

Abstract

We consider a hybrid FEM-BEM method to compute approximations of full-space linear elliptic transmission problems. First, we derive a priori and a posteriori error estimates. Then, building on the latter, we present an adaptive algorithm and prove that it converges at optimal rates with respect to the number of mesh elements. Finally, we provide numerical experiments, demonstrating the practical performance of the adaptive algorithm.

Figures (7)

• Figure 6.1: Domains and initial meshes $\mathcal{T}_0$.
• Figure 6.2: Experiments in Section \ref{['sec:numerics_square']}: Plots of the approximation $u_4$ and of its components $u_{1,4}$ and $u_{2,4}$.
• Figure 6.3: Experiments in Section \ref{['sec:numerics_square']}: Error $\|u - u_\ell\|_{H^1(\Omega)}$ and error estimates $\eta_\ell$, $\eta_{1,\ell}$, and $\eta_{2,\ell}$ plotted against the total number of degrees of freedom $N_\ell$. Comparison of uniform and adaptive mesh refinement.
• Figure 6.4: Experiments in Section \ref{['sec:numerics_Lshaped']}: Plots of the approximation $u_{14}$ and of its components $u_{1,{14}}$ and $u_{2,{14}}$.
• Figure 6.5: Experiments in Section \ref{['sec:numerics_Lshaped']}: Error $\|u - u_\ell\|_{H^1(\Omega)}$ and error estimates $\eta_\ell$, $\eta_{1,\ell}$, and $\eta_{2,\ell}$ plotted against the total number of degrees of freedom $N_\ell$. Comparison of uniform and adaptive mesh refinement.

Theorems & Definitions (10)

• Remark 3.1
• Remark 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5
• proof
• Theorem 4.2
• proof
• Remark 4.3