NDLI: The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	LeVeque, Randall J. Li, Zhilin
Copyright Year	1994
Abstract	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 Peskins immersed boundary method.
Starting Page	1019
Ending Page	1044
Page Count	26
File Format	PDF
ISSN	00361429
DOI	10.1137/0731054
e-ISSN	10957170
Issue Number	4
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-14
Access Restriction	Subscribed
Subject Keyword	singular source term delta functions discontinuous coefficients finite difference methods Mesh generation and refinement irregular domain interface elliptic equation Finite difference methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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