Additive Schwarz method for the p -version of the boundary element method for the single layer potential operator on a plane screen

Abstract

We analyze an additive Schwarz preconditioner for the p-version of the boundary element method for the single layer potential operator on a plane screen in the three-dimensional Euclidean space. We decompose the ansatz space, which consists of piecewise polynomials of degree p on a mesh of size h, by introducing a coarse mesh of size $H\ge h$ . After subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces of piecewise polynomials living only on elements of size H. This decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by $O(\log \frac Hhp)^2$ . Numerical results supporting the theory are presented.