1996•SIAM Journal on Numerical AnalysisRequires access

A Uniformly Convergent Finite Difference Scheme for a Singularly Perturbed Semilinear Equation

Paul A. Farrell, John J. H. Miller, Eugene O’Riordan, Gregorii I. Shishkin

Abstract

Boundary value problems for singularly perturbed semilinear elliptic equations are considered. Special piecewise-uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter. Numerical methods composed of standard monotone finite difference operators and these piecewise-uniform meshes are shown theoretically to be uniformly (with respect to the singular perturbation parameter) convergent. Numerical results are also presented, which indicate that in practice the method is first-order accurate.

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Key concepts: Mathematics, Singular perturbation, Uniform convergence, Piecewise, Polygon mesh, Monotone polygon, Mathematical analysis, Perturbation (astronomy)