Domain decomposition methods for boundary integral equations of the first kind: numerical results

Abstract

We present numerical experiments for the additive Schwarz algorithms applied to the h- and p-version Galerkin boundary element methods to solve the Laplacian and Helmholtz boundary value problems in two dimensions. Both weakly singular and hypersingular integral equations covering Dirichlet and Neumann problems, respectively, are considered. In the case of Laplacian problems where the Galerkin scheme yields symmetric and positive definite stiffness matrices, we use the preconditioned CG method, whereas in the case of Helmholtz problems we have indefinite non Hermitian stiffness matrices, and therefore we use the preconditioned GMRES method. We find that the two level additive Schwarz methods yield only logarithmically growing condition numbers for both the h- and p-versions; thus only a fixed number of iterations is necessary to compute appropriate approximations of the Galerkin solutions. We also perform multilevel methods for integral equations belonging to the boundary value problems arising from the Laplacian and the Helmholtz equation. Here we observe (for both the weakly singular and hypersingular integral equations) that the condition numbers of the preconditioned systems grow like plog³p for the p-version. For the h-version we find in the case of the weakly singular equation (with the use of the Haar basis in the construction of the preconditioner) mildly increasing condition numbers, whereas for the hypersingular equation the multilevel additive Schwarz operator has bounded condition number and gives just the BPX preconditioner.

Keywords:

• domain decomposition
• additive schwarz
• boundary element method
• conjugate gradient
• GMRES