A Penalty-Free SGFEM for Interface Problems With Nonhomogeneous Jump Conditions

Abstract

In this paper, we present a stable generalized finite element method (SGFEM) to address the approximation of the discontinuous solutions of interface problems with nonhomogeneous interface conditions. We propose a set of enrichment functions based on the Heaviside and Distance functions on the patches that intersect the interface. The enrichment based on the Heaviside function is used to strongly enforce the given nonhomogeneous interface condition, that is, the jump in the solution, and only the enrichment based on the product of the Heaviside and Distance functions contributes to the degrees of freedom. Consequently, the number of degrees of freedom in this approach is the same as that is required for an interface problem with homogeneous interface conditions. The chief merit is that the proposed method totally confors and does not use conventional techniques for the nonhomogeneous interface condition in the literature, such as the penalty method or the Lagrange multiplier. Our experiments show that this method yields optimal order of convergence, its conditioning is not worse than that of the standard finite element method, and it is robust.