**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", }