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Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem

  • Natalia Kopteva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)

Abstract

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.

Keywords

Mesh Node Polygonal Domain Singularly Perturb Shishkin Mesh Approximate Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Natalia Kopteva
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of LimerickLimerickIreland

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