Multilevel methods for the h-, p-, and hp-versions of the boundary element method

In this paper we give an overview on the definition of finite element spaces for the h-, p-, and hp-version of the BEM along with preconditioners of additive Schwarz type. We consider screen problems (with a hypersingular or a weakly singular integral equation of first kind on an open surface Γ) as model problems. For the hypersingular integral equation and the h-version with piecewise bilinear functions on a coarse and a fine grid we analyze a preconditioner by iterative substructuring based on a non-overlapping decomposition of Γ. We prove that the condition number of the preconditioned linear system behaves polylogarithmically in H/h. Here H is the size of the subdomains and h is the size of the elements. For the hp-version and the hypersingular integral equation we comment in detail on an additive Schwarz preconditioner which uses piecewise polynomials of high degree on the fine grid and yields also a polylogarithmically growing condition number. For the weakly singular integral equation, where no continuity of test and trial functions across the element boundaries has to been enforced, the method works for nonuniform degree distributions as well. Numerical results supporting our theory are reported.