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 ∇⋅(β(x)∇u(x))+κ(x)u(x)=f(x) in a region Ω in 1 or 2 space dimensions. We assume that Ω 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 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. We also allow the specification of a jump discontinuity in u itself across Γ. We show 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 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", }