Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems

Penalty combination of the Ritz-Galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities. The superconvergence rate, O(h2−δ), of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids, where h is the maximal mesh length of difference grids used in the finite difference method, and δ(> 0) is an arbitrarily small number. It is due to its simplicity that the penalty combination of the Ritz-Galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces.