THE IMMERSED INTERFACE METHOD FOR ELLIPTIC-EQUATIONS WITH DISCONTINUOUS COEFFICIENTS AND SINGULAR SOURCES

The authors develop finite difference methods for elliptic equations of the form del . (beta(x)delu(x)) + kappa(x)u(x) = f(x) in a region OMEGA in one or two space dimensions. It is assumed that OMEGA is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in which there is an irregular surface GAMMA of codimension 1 contained in OMEGA across which beta, kappa, and f may be discontinuous, and along which the source f may have a delta function singularity. As a result. derivatives of the solution u may be discontinuous across GAMMA. The specification of a jump discontinuity in u itself across GAMMA is allowed. It is shown that it is possible to modify the standard centered difference approximation to maintain second order accuracy on the uniform grid even when GAMMA is not aligned with the grid. This approach is also compared with a discrete delta function approach to handling singular sources, as used in Peskin's immersed boundary method.