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.

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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",
}

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