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Adaptive coupling of boundary elements and finite elements. (English) Zbl 0849.65083
The authors consider the interface problem (1)–(4) defined by the (possibly nonlinear) partial differential equation (1) \(-\text{div}(A\text{ grad}(u))= f\) in a plane, bounded Lipschitz domain \(\Omega\) with the boundary \(\Gamma\), by the Laplace equation (2) \(- \Delta v= 0\) in the complement \(\Omega^c:=\mathbb{R}^2\backslash \overline\Omega\), by the radiation condition (3) \(v(x)= (b/2\pi) \log|x|+ o(1)\) as \(|x|\to \infty\), and, finally, by the prescribed jump conditions (4) \(u= v+ u_0\) and \((A\text{ grad}(u)) n= \partial v/\partial n+ t_0\) on the interface \(\Gamma\), with given \((f, u_0, t_0)\in L^2(\Omega)\times H^{1/2}(\Gamma)\times H^{- 1/2}(\Gamma)\).
Using the Cauchy data representation on \(\Gamma\) via boundary integral operators, the authors arrive at a symmetric coupled boundary and domain integral variational reformulation of the interface problem (1)–(4) which is the starting point for the Galerkin discretization. Existence and uniqueness of the solutions as well as Cea-type discretization error estimates in the “energy” norm are standard.
The main result of the paper is an a posteriori discretization error estimate in the “energy” norm providing a computable upper bound up to a multiplicative constant. This bound can be used for a residual based adaptive discretization procedure as it is known from the finite element method. The numerical test examples (L- and Z-shapes) confirm that the procedure proposed produces reasonable meshes.
Reviewer: U.Langer (Linz)

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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