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A Posteriori and a Priori Techniques of Local Grid Refinement for Parabolic Problems with Boundary and Transition Layers

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Numerical Analysis and Its Applications (NAA 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1988))

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Abstract

A Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation is considered on a segment and, in particular, in a composite domain. The solution of such a problem exhibits boundary and transition (in the case of the composite domain) parabolic layers. For this problem we study classical difference approximations on sequentially locally refined meshes. The correction of the discrete solutions is performed only on the subdomains subjected to refinement (their boundaries pass through the grid nodes); uniform meshes are used in these adaptation subdomains. For a posteriori grid refinement we apply, as indicators, auxiliary functions majorizing the singular component of the solution. As was shown, in this class of the finite difference schemes there exist no schemes which converge independently of the singular perturbation parameter ε (or ε-uniformly). We construct special schemes, which allow us to obtain the approximations that converge “almost ε- uniformly”, i.e., with an error weakly depending on ε.

This research was supported in part by the Russian Foundation for Basic Research under grant No. 98-01-00362 and by the NWO grant dossiernr. 047.008.007.

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Shishkin, G.I. (2001). A Posteriori and a Priori Techniques of Local Grid Refinement for Parabolic Problems with Boundary and Transition Layers. In: Vulkov, L., Yalamov, P., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2000. Lecture Notes in Computer Science, vol 1988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45262-1_84

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  • DOI: https://doi.org/10.1007/3-540-45262-1_84

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41814-6

  • Online ISBN: 978-3-540-45262-1

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