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A linear Uzawa-type FEM-BEM solver for nonlinear transmission problems. (English) Zbl 1415.65247

Summary: We propose a fully discrete Uzawa-type iteration for the Johnson-Nédélec formulation of a Laplace-type transmission problem with possible (strongly monotone) nonlinearity in the interior domain. In each step, we sequentially solve one BEM for the weakly-singular integral equation associated with the Laplace-operator and one FEM for the linear Yukawa equation. In particular, the nonlinearity is only evaluated to build the right-hand side of the Yukawa equation. The algorithm includes the inexact solution of the BEM/FEM part by a preconditioned CG method. We prove that the proposed method leads to linear convergence with respect to the number of Uzawa iterations. Moreover, while the current analysis of a direct FEM-BEM discretization of the Johnson-Nédélec formulation requires some restrictions on the ellipticity (resp. strong monotonicity constant) in the interior domain, our Uzawa-type solver avoids such assumptions.

MSC:

65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J62 Quasilinear elliptic equations
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