Augmented immersed finite element methods for some elliptic partial differential equations

Abstract

Augmented immersed finite element methods are proposed to solve elliptic interface problems with non-homogeneous jump conditions. The non-homogeneous jump conditions are treated as source terms using the singularity removal technique. For the piecewise constant coefficient case, we transform the original interface problem to a Poisson equation with the same jump in the solution, but an unknown flux jump (augmented variable) which is chosen such that the original flux jump condition is satisfied. The GMRES iterative method is used to solve the augmented variable. The core of each iteration involves solving a Poisson equation using a fast Poisson solver and an interpolation scheme to interpolate the flux jump condition. With a little modification, the method can be applied to solve Poisson equations on irregular domains. Numerical experiments show that not only the computed solution but also the normal derivative are second-order accurate in the norm.