Hierarchical Basis Preconditioners for Coupled Fem{bem Equations

SUMMARY The purpose of this paper is to present a nearly optimal preconditioned iterative method to solve indeenite linear systems of equations arising from h-adaptive procedures for the symmetric coupling of Finite Elements and Boundary Elements. This solver is nearly optimal in the sense, that its convergence rate grows only logarithmically with the number of unknowns. The algorithm is based on the conjugate residual method with block-diagonal pre-conditioning, where no Schur complement construction is required. This method uses diierent hierarchical basis preconditioners for the positive semi{deenite FEM block belonging to an interior Neumann problem and the negative deenite boundary element block belonging to the single layer potential. The eeciency of the hierarchical basis solvers is underlined by a numerical experiment showing fast convergence.