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An accurate semi‐analytic finite difference scheme for two‐dimensional elliptic problems with singularities
Author(s) -
Yosibash Z.,
Arad M.,
Yakhot A.,
BenDor G.
Publication year - 1998
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199805)14:3<281::aid-num1>3.0.co;2-i
Subject(s) - mathematics , gravitational singularity , finite difference , mathematical analysis , finite difference method , domain (mathematical analysis) , rate of convergence , boundary value problem , finite difference scheme , partial differential equation , channel (broadcasting) , electrical engineering , engineering
A high‐order semi‐analytic finite difference scheme is presented to overcome degradation of numerical performance when applied to two‐dimensional elliptic problems containing singular points. The scheme, called Least‐Square Singular Finite Difference Scheme (L‐S SFDS), applies an explicit functional representation of the exact solution in the vicinity of the singularities, and a conventional finite difference scheme on the remaining domain. It is shown that the L‐S SFDS is “pollution” free, i.e., no degradation in the convergence rate occurs because of the singularities, and the coefficients of the asymptotic solution in the vicinity of the singularities are computed as a by‐product with a very high accuracy. Numerical examples for the Laplace and Poisson equations over domains containing re‐entrant corners or abrupt changes in the boundary conditions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 281–296, 1998