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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem
verfasst von : Natalia Kopteva
Erschienen in: Numerical Analysis and Its Applications
Verlag: Springer Berlin Heidelberg
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A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter
ε
2
is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Math. Comp. 76 (2007) 631–646], in which a parametrization of the boundary
$\partial\Omega$
is assumed to be known, to a more practical case when the domain is defined by an ordered set of boundary points. It is shown that, using layer-adapted meshes, one gets second-order convergence in the discrete maximum norm, uniformly in
ε
for
ε
≤
Ch
. Here
h
> 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed
Ch
− 2
. Numerical results are presented that support our theoretical error estimates.