The Explicit Jump Immersed Interface Method: Finite Difference Methods for Pde with Piecewise Smooth Solutions We Dedicate This Work to Helmut Rohrl on the Occasion of His 70th Birthday

Many boundary value problems (BVPs) or initial BVPs have non-smooth solutions, with jumps along lower-dimensional interfaces. The Explicit{Jump Immersed Interface Method (EJIIM) was developed following Li's Fast Iterative IIM (FIIIM), recognizing that the foundation for the eecient 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 coeecients and extends the IIM to BVPs on irregular domains with Neumann and Dirichlet boundary conditions. A proof of second order convergence for problems with singular sources is 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 eeciency of the FIIIM. 1. Introduction Standard nite diierence approximations fail when applied to non-smooth functions because the Taylor expansions that they are based on are not valid. But many applications lead to non-smooth solutions. These can be dealt with by adaptive methods, for example by nite element methods, where element boundaries may be chosen to coincide with the interface, or by boundary tted nite diierence schemes. Considerable eeort in constructing a grid has to be spent in these cases. We concern ourselves with problems where this grid construction is not aaordable, presumably because the location of the discontinuity is either not xed (for example in moving interface problems), or to be determined (as in inverse problems or in design problems). The discontinuities are assumed