Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity
For researchers in computational mechanics and numerical analysis, this work offers rigorous theoretical foundations and adaptive algorithms for solving nonlinear interface problems with FEM-BEM coupling.
This paper provides a unified framework for analyzing three classical FEM-BEM coupling methods for nonlinear interface problems, proving well-posedness, quasi-optimality, and convergence of adaptive mesh refinement via residual-based error estimators. Numerical experiments compare the methods' performance on problems with singularities.
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-Nédélec 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 Céa-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.