Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem
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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.
KeywordsMesh Node Polygonal Domain Singularly Perturb Shishkin Mesh Approximate Curvature
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- 3.Grindrod, P.: Patterns and Waves: the Theory and Applications of Reaction-Diffusion Equations. Clarendon Press (1991)Google Scholar
- 5.Kopteva, N.: Pointwise error estimates for 2nd singularly perturbed semilinear reaction-diffusion problems. In: Farago, I., Vabishchevich, P., Vulkov, L. (eds.) Finite Difference Methods: Theory and Applications. Proceedings of the 4th International Conference, Lozenetz, Bulgaria, pp. 105–114 (2006)Google Scholar
- 9.Samarski, A.A.: Theory of Difference Schemes. Nauka (1989) (in Russian)Google Scholar
- 10.Shishkin, G.I.: Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Ur. O. Ran, Ekaterinburg (1992) (in Russian)Google Scholar