Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary.

We study some additive Schwarz algorithms for the $p$ version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as $(1+\log p)^2$ independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators.