Modified intrinsic extended finite element method for elliptic equation with interfaces

Abstract

In this paper, a modified intrinsic extended finite element method (XFEM) for one-dimensional and two-dimensional elliptic equations with discontinuous coefficients and interfaces is proposed. We improve the intrinsic XFEM by changing the shape functions of the critical nodes. The improved shape functions can be used to catch the discontinuous information near interfaces. In addition, we modify the Gauss integration in special elements cut by interfaces. Numerical experiments are presented to verify the feasibility and superiority of the modified intrinsic XFEM compared with the standard FEM and extrinsic XFEM for this type of problem. Results also show that the modified intrinsic XFEM can generate an approximate solution whose error is $$O(h^2)$$ in an $$\hbox {L}^2$$ -norm and O(h) in an energy norm if the Q1 element is used.