A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations
type: Item
instance of: scholarly article
title: A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations
zbMATH Open document ID: 1185.65217
DOI: 10.1504/IJCSM.2007.016542
author: G. I. Shishkin
published in: International Journal of Computing Science and Mathematics
publication date: 2010-05-05T00:00:00Z
review text: Summary: Dirichlet problem is considered for a singularly perturbed semilinear parabolic convection-diffusion equation on a rectangular domain. The solution of the classical finite difference scheme on a uniform mesh converges at the rate \(\mathcal O((\varepsilon + N^{-1})^{-1} N^{-1} + N_0^{-1})\) where \(N + 1\) and \(N_0 + 1\) denote the numbers of mesh points with respect to \(x\) and \(t\) respectively, \(\varepsilon \in (0,1]\) is the perturbation parameter. Using nonlinear and linearised basic classical schemes, finite difference schemes on a posteriori adaptive meshes based on uniform subgrids are constructed. The subdomains where grid refinement is required are defined by the gradients of the solutions of the intermediate discrete problems. The constructed difference schemes converge 'almost \(\varepsilon \)-uniformly', namely, at the rate \(\mathcal O((\varepsilon ^{-\nu} N^{-1}) + N^{-1/2} + N_0^{-1})\) where \(\nu \) is an arbitrary number from \((0, 1]\).
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 5703887
zbMATH Keywords: singular perturbations
zbMATH Keywords: boundary value problems
zbMATH Keywords: semilinear parabolic convection-diffusion equations
zbMATH Keywords: finite difference approximation
zbMATH Keywords: a posteriori adaptive meshes
zbMATH Keywords: uniform subgrids
zbMATH Keywords: almost \(\varepsilon \)-uniform convergence
zbMATH Keywords: gradient of the discrete solution
MaRDI profile type: Publication
full work available at URL: ijcsm.2007.016542
OpenAlex ID: W2114375925
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description: scientific article; zbMATH DE number 5703887