Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

For the simple layer potential V associated with the three-dimensional (3D) Laplacian, we consider the weakly singular integral equation Vφ=f. This equation is discretized by the lowest-order Galerkin boundary element method. We prove convergence of an h-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorithm converges quasi-optimally with respect to the number of elements. In particular, we prove that adaptive mesh refinement is superior to uniform mesh refinement.