Optimal Quadratic Nitsche Extended Finite Element Method for Interface Problem of Diffusion Equation

Abstract

In this paper, we study Nitsche extended finite element method (XFEM) for the interface problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular family of grids and prove the optimal convergence rate of the scheme with respect to the mesh size. Main efforts are devoted onto classifying the cases of intersection between the elements and the interface and prove a weighted trace inequality for the extended finite element functions needed, and the general framework of analysing XFEM can be implemented then.