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

Abstract Many boundary value problems BVPs or initial BVPs have nonsmooth solutions with jumps along lowerdimensional interfaces The ExplicitJump Immersed Interface Method EJIIM was developed following Lis Fast Iterative IIM FIIIM recognizing that the foundation for the e cient 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 coe cients 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 e ciency of the FIII