An augmented immersed finite element method for variable coefficient elliptic interface problems in two and three dimensions

Abstract

An augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients. By introducing the jump of normal derivative of the solution along the interface as an augmented variable, the interface problem is solved iteratively on Cartesian meshes using the generalized minimal residual (GMRES) iterative method. Within each iteration, we need to solve a system of linear equations with the same symmetric and positive definite matrix but with different right-hand sides. The sequence of linear equations is solved efficiently by the multi-grid solvers since the set up of coarse grid matrices and decompositions is only computed once. Extensive numerical examples show that the number of GMRES iterations is nearly independent of the mesh size when a diagonal-like preconditioning is applied. Thus the computational work of the proposed method is linearly proportional to the number of grid nodes. Implementation details are given and extensive numerical examples in two and three dimensions are provided to show the accuracy and efficiency of the proposed method.