The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
by R. J. LeVeque and Z. Li SIAM J. Numer. Anal., 31(1994), pp. 1019-1044

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",
}