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

Randall J. Leveque and Zhilin Li
SIAM Journal on Numerical Analysis
Vol. 31, No. 4 (Aug., 1994), pp. 1019-1044
Stable URL: http://www.jstor.org/stable/2158113
Page Count: 26
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The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
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Abstract

The authors develop finite difference methods for elliptic equations of the form $\nabla \centerdot (\beta(x)\nabla u(x)) + \kappa(x)u(x) = f(x)$ in a region Ω in one or two space dimensions. It is assumed that Ω 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 Γ of codimension 1 contained in Ω across which β, κ, 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 Γ. The specification of a jump discontinuity in u itself across Γ 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 Γ 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.

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