The hp-version of the boundary element method with quasi-uniform meshes for weakly singular operators on surfaces

We prove an a priori error estimate for the hp-version of the boundary element method with weakly singular operators in three dimensions. The underlying meshes are quasi-uniform. Our model problem is that of the Laplacian exterior to an open surface, where the solution has strong singularities that are not L2-regular. Our results confirm previously conjectured convergence rates in h (the mesh size) and p (the polynomial degree) and these rates are given explicitly in terms of the exponents of the singular functions. In particular, for sufficiently smooth given data we prove a convergence in the energy norm like O(h 1/2p -1).