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

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 original 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.

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