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 \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] 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.