Hierarchical preconditioners for boundary element domain decomposition methods

\underline{O. Steinbach}, W. L. Wendland

Universität Stuttgart, Mathematisches Institut A Pfaffenwaldring 57, D–70569 Stuttgart

             E–mail: steinbach/wendland@mathematik.uni-stuttgart.de

Abstract

For the solution of mixed boundary value problems we use explicit representations of the Steklov– Poincaré operator and its inverse. Based on these relations one can construct preconditioners for the local single layer potentials as well as for the Schur complement system associated with the global problem. These preconditioners yield spectral equivalence inequalities which are independent of the actual discretization parameters, if some appropriate conditions for the trial and test spaces are satisfied. However, in domain decomposition methods involving inner coarse grid nodes one has to introduce an additional coarse grid system, which can be generated by local Trefftz elements for arbitrary subdomains in terms of local boundary integral equations. This approach allows to construct different preconditioned iterative algorithms, which are suitable for the symmetric as well as for the non–symmetric discretizations based on Galerkin as well as on collocation formulations, and moreover, for the corresponding Schur complement system as well as for the coupled system. Numerical examples for the Laplacian and for problems in linear elasticity will be given.