The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions

Many boundary value problems (BVPs) or initial BVPs have nonsmooth solutions, with jumps along lower-dimensional interfaces. The explicit-jump immersed interface method (EJIIM) was developed following Li's fast iterative immersed interface method (FIIIM), recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic BVPs. EJIIM generalizes the class of problems for which FIIIM is applicable. It handles interfaces between constant and variable coefficients and extends the immersed interface method (IIM) to BVPs on irregular domains with Neumann and Dirichlet boundary conditions. Proofs of second order convergence for a one-dimensional (1D) problem with piecewise constant coefficients and for two-dimensional (2D) problems with singular sources are given. Other problems are reduced to the singular sources case, with additional equations determining the source strengths. The advantages of EJIIM are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries, while still maintaining the efficiency of the FIIIM.