Multilevel Additive Schwarz Method for the $h$-$p$ Version of the Galerkin Boundary Element Method

We study a multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the $h$-$p$ version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns $M$. We prove that the condition number $\kappa(P)$ of the multilevel additive Schwarz operator behaves like $O(\sqrt{M}\log^2M)$. As a direct consequence of this we also give the results for the $2$-level preconditioner and also for the $h$-$p$ version with quasi-uniform meshes. Numerical results supporting our theory are presented.