**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", }