Abstract. We develop finite difference methods for elliptic equations of the form \[ \nabla\cdot(\beta(x)\nabla u(x)) + \kappa(x)u(x)=f(x) \] in a region \(\Omega\) in 1 or 2 space dimensions. We assume that \(\Omega\) is a simple region (e.g., a rectangle) and that we wish to use a uniform rectangular grid. We study the situation 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 \(\Gamma\). We also allow the specification of a jump discontinuity in \(u\) itself across \(\Gamma\). We show 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 compared with a discrete delta function approach to handling singular sources, as used in Peskin's immersed boundary method.
bibtex entry:
@Article{rjl-li:sinum,
author = "R. J. LeVeque and Z. Li",
title = "The Immersed Interface Method for Elliptic Equations
with Discontinuous Coefficients and Singular Sources",
journal = "SIAM J. Numer. Anal.",
volume = "31",
pages = "1019--1044",
year = "1994",
}