The immersed interface/multigrid methods for interface problems

New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.