The singular finite element method for some elliptic boundary value problem with interface

A kind of numerical method is proposed for some elliptic boundary value problems with interface. It includes mainly two steps. At the first step, we obtain the approximation of the singularity near the singular points by solving a simple eigenvalue problem, which is one dimension less than the original problem. At the second step, we apply the approximation of the singularity together with the standard finite element basis functions to construct the singular finite element space and finally solve the original problem on a conventional mesh. Some numerical examples show the effectiveness of this method.