The Immersed Weak Galerkin and Continuous Galerkin Finite Element Method for Elliptic Interface Problem

Abstract: In this paper, we use the weak Galerkin finite element method to solve the elliptic interface problem on interface-independent meshes. In the interface element, we use the immersed finite element (IFE) functions satisfying the interface conditions precisely and they have optimal approximation capabilities. In the non-interface element, the continuous element is employed to approximate the exact solution. The optimal convergence orders of error are obtained in the $H^1$ norm and $L^2$ norm. A series of numerical experiments are provided to validate the efficiency of the proposed method.