Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

TL;DR

This paper develops a unified framework for nonlinear FEM-BEM coupling methods in 2D and 3D, proving well-posedness, convergence, and effectiveness of adaptive strategies with novel inverse estimates and numerical validation.

Contribution

It introduces a comprehensive analysis of nonlinear FEM-BEM couplings, establishing well-posedness, quasi-optimality, and convergence of adaptive algorithms with new inverse boundary operator estimates.

Findings

Proved well-posedness of nonlinear FEM-BEM coupling solutions.

Established convergence of adaptive FEM-BEM methods.

Validated effectiveness through numerical experiments with singularities.

Abstract

We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson-N\'ed\'elec coupling, the Bielak-MacCamy coupling, and Costabel's symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a C\'ea-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.