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Caccioppoli-type estimates and $\mathcal{H}$-Matrix approximations to inverses for FEM-BEM couplings

Markus Faustmann, Jens Markus Melenk, Maryam Parvizi

TL;DR

This work analyzes three FEM-BEM couplings for transmission problems on Lipschitz domains, deriving discrete Caccioppoli-type interior regularity estimates for the Bielak–MacCamy, Costabel symmetric, and Johnson–Nédélec formulations. Leveraging these regularity results, the authors develop an abstract framework to approximate inverses in the $\mathcal{H}$-matrix format and prove exponential convergence of the inverses on admissible blocks for the lowest-order discretizations. Theoretical results are complemented by a detailed numerical study that confirms exponential convergence with respect to block rank and demonstrates the effectiveness of block-diagonal $\mathcal{H}$-LU preconditioning for large FEM-BEM systems. The findings provide a rigorous basis for efficient, data-sparse stiffness-matrix inverses and practical preconditioning strategies in FEM-BEM couplings.

Abstract

We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak-MacCamy coupling, the symmetric coupling, and the Johnson-Nédélec coupling. For each coupling we provide discrete interior regularity estimates. As a consequence, we are able to prove the existence of exponentially convergent $\mathcal{H}$-matrix approximants to the inverse matrices corresponding to the lowest order Galerkin discretizations of the couplings.

Caccioppoli-type estimates and $\mathcal{H}$-Matrix approximations to inverses for FEM-BEM couplings

TL;DR

This work analyzes three FEM-BEM couplings for transmission problems on Lipschitz domains, deriving discrete Caccioppoli-type interior regularity estimates for the Bielak–MacCamy, Costabel symmetric, and Johnson–Nédélec formulations. Leveraging these regularity results, the authors develop an abstract framework to approximate inverses in the -matrix format and prove exponential convergence of the inverses on admissible blocks for the lowest-order discretizations. Theoretical results are complemented by a detailed numerical study that confirms exponential convergence with respect to block rank and demonstrates the effectiveness of block-diagonal -LU preconditioning for large FEM-BEM systems. The findings provide a rigorous basis for efficient, data-sparse stiffness-matrix inverses and practical preconditioning strategies in FEM-BEM couplings.

Abstract

We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak-MacCamy coupling, the symmetric coupling, and the Johnson-Nédélec coupling. For each coupling we provide discrete interior regularity estimates. As a consequence, we are able to prove the existence of exponentially convergent -matrix approximants to the inverse matrices corresponding to the lowest order Galerkin discretizations of the couplings.

Paper Structure

This paper contains 20 sections, 13 theorems, 157 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Bielak–MacCamy coupling
  4. Costabel's symmetric coupling
  5. The Johnson-Nédélec coupling
  6. $\mathcal{H}$-Matrix approximation of inverses
  7. The Caccioppoli-type inequalities
  8. The Bielak-MacCamy coupling
  9. The symmetric coupling
  10. The Johnson-Nédélec coupling
  11. $\mathcal{H}$-matrix approximation to inverse matrices
  12. Abstract setting - from matrices to functions
  13. Abstract setting - low dimensional approximation
  14. Application of the abstract framework for the FEM-BEM couplings
  15. The local dual basis
  16. ...and 5 more sections

Key Result

Theorem 2.3

Assume that $C_{\rm ell} > 1/4$ in (eq:Cell). Let $\varepsilon \in (0,1)$ and $R \in (0,2\operatorname*{diam}(\Omega))$ be such that $\frac{h}{R} < \frac{\varepsilon}{16}$, and let $B_R$ and $B_{(1+\varepsilon)R}$ be two concentric boxes. Assume that the data is localized away from $B_{(1+\varepsilo where the norms on the right-hand side are defined in eq:def:triplenorm.

Figures (1)

  • Figure 1: $\mathcal{H}$-matrix approximation to inverse FEM-BEM matrix; left: error vs. block rank $r$; right: memory requirement vs. block rank $r$; top: $N= 6959$ (FEM-dofs), $M=3888$ (BEM-dofs); bottom: $N=10648$, $M=5292$.

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6: bounding boxes and $\eta$-admissibility
  • Remark 2.7
  • Definition 2.8: cluster tree
  • Definition 2.9: far field, near field, and sparsity constant
  • Definition 2.10: $\mathcal{H}$-matrices
  • ...and 25 more