A generalized finite difference method for solving elliptic interface problems with non-homogeneous jump conditions on surfaces

Abstract

A meshless generalized finite difference method is presented to solve elliptic interface problems with non-homogeneous jump conditions on surfaces. The innovation is that based on a splitting treatment of the interface problem, the generalized finite difference method combined with the local tangential lifting approach is developed for the second-order spatial discretization with low complexity and small stencil. The generalized finite difference method considered is based on Taylor expansion and weighted moving least squares approximation. Under the local tangential lifting framework, the numerical scheme formed by the generalized finite difference method on the tangent plane is regarded as that for the original surface problem. The biggest advantage for combining two methods is that the local tangential lifting framework can reduce the complexity of the discrete process of generalized finite difference method on the surface and resulting relatively small stencil. Several numerical examples are presented to validate the convergence of the proposed method. Numerical results suggest that the proposed approach is effective to the surface interface problem with non-homogeneous jump conditions.