Adaptive boundary element methods for some first kind integral equations

In this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the Dirichlet problem and the Neumann problem for the Laplacian in a two-dimensional Lipschitz domain. For the h-version of the finite element Galerkin discretization of the single layer potential and the hypersingular operator, we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm (up to a multiplicative constant). Following Eriksson and Johnson this yields adaptive algorithms steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate.