Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh-refinement

摘要

We consider the adaptive lowest-order boundary element method based on isotropic mesh refinement for the weakly-singular integral equation for the three-dimensional Laplacian. The proposed scheme resolves both, possible singularities of the solution as well as of the given data. The implementation thus only deals with discrete integral operators, that is, matrices. We prove that the usual adaptive mesh-refining algorithm drives the corresponding error estimator to zero. Under an appropriate saturation assumption which is observed empirically, the sequence of discrete solutions thus tends to the exact solution within the energy norm.