The $hp$-Version of BEM — Fast Convergence, Adaptivity and Efficient Preconditioning

Abstract

In this survey paper we report on recent developments of the $hp$-version of the boundary element method (BEM). As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar, open surface. We show that the Galerkin solutions (computed with the $hp$-version on geometric meshes) converge exponentially fast towards the exact solutions of the integral equations. An $hp$-adaptive algorithm is given and the implementation of the $hp$-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices. We also comment on the use of the $hp$-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points. Numerical results are presented which underline the theoretical results.