A fast enriched FEM for Poisson equations involving interfaces

Abstract

We develop a fast enriched finite element method for solving Poisson equations involving complex geometry interfaces by using regular Cartesian grids. The presence of interfaces is accounted for by developing suitable jump conditions. The immersed boundary method (IBM) and the immersed interface method (IIM) are successfully used to solve these problems when combined with a fast Fourier transform. However, the IBM and the IIM, which are developed from the finite difference method, have several disadvantages including the characterization of the null spaces and the inability to treat complex geometries accurately. We propose a solution to these difficulties by employing the finite element method. The continuous Galerkin solution approximations at the interface elements are modified using the enriched basis functions to make sure that the optimal convergence rates are obtained. Here, the FFT is applied in the fast Poisson solver to significantly accelerate the computational processes for solving the global matrix system. With reasonably small interfaces, the operational cost is almost linearly proportional to the number of the Cartesian grid points. The method is further extended to solve problems involving multi-materials while preserving the optimal accuracy. Several benchmark examples are shown to demonstrate the performance of the method.