Maximum norm error analysis of a 2D singularly perturbed semilinear reaction-diffusion problem

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Abstract

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) 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 experiments are performed to support the theoretical results.

Original languageEnglish
Pages (from-to)631-646
Number of pages16
JournalMathematics of Computation
Volume76
Issue number258
DOIs
Publication statusPublished - Apr 2007

Keywords

  • Bakhvalov mesh
  • Maximum norm error estimate
  • Second order
  • Semilinear reaction-diffusion
  • Shishkin mesh
  • Singular perturbation
  • Z-field

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