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We construct a reliable and efficient residual-based local a posteriori error estimator for a Galerkin method coupling finite elements and boundary elements for an eddy current problem in a three-dimensional polyhedral domain. For the proof of the efficiency of the error estimator, we assume that the boundary mesh is quasi-uniform and that the boundary surface and the boundary data satisfy certain smoothness assumptions. The Galerkin method uses lowestorder Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise bilinear functions on the boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in terms of the error estimator as well. The estimator is derived from the defect equation using a Helmholtz decomposition and Green's formulas. The decomposed parts of the Galerkin error are approximated by local interpolation operators. Numerical tests underline reliability and efficiency of the residual error estimator.

We study an additive Schwarz based preconditioner for the hp-version of the boundary element method with quasi-uniform triangular meshes and for hypersingular integral operators. The model problem is Laplace's equation exterior to an open surface and is generic for elliptic boundary value problems of second order in bounded and unbounded domains with closed or open boundary. The preconditioner is based on a nonoverlapping subspace decomposition into a so-called wire basket space and interior functions for each element. We prove that the condition number of the preconditioned stiffness matrix has a bound, which is independent of the mesh size h and which grows only polylogarithmically in p, the maximum polynomial degree. Numerical experiments confirm this result.

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We present an hp-version of the finite element / boundary element coupling method to solve the eddy current problem for the time-harmonic Maxwell's equations. We use H(curl, )-conforming vector-valued polynomials to approximate the electric field in the conductor and surface curls of continuous piecewise polynomials on the boundary of to approximate the twisted tangential trace of the magnetic field on . We present both a priori and a posteriori error estimates together with a three-fold hp-adaptive algorithm to compute the fem/bem coupling solution with appropriate distributions of polynomial degrees on suitably refined meshes.