A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions

A second order accurate finite difference method is presented for solving two-dimensional variable coefficient elliptic equations on Cartesian grids, in which the coefficients, the source term, the solution and its derivatives may be nonsmooth or discontinuous across an interface. A correction term is introduced to the standard central difference stencil so that the numerical discretization is well-defined across the interface. We also propose a new method to approximate the correction term as part of the iterative procedure. The method is easy to implement since the correction term only needs to be added to the right-hand-side of the system. Therefore, the coefficient matrix remains symmetric and diagonally dominant, allowing for most standard solvers to be used. Numerical examples show good agreements with exact solutions, and the order of accuracy is comparable with other immersed interface methods.