Analysis of a Galerkin Finite Element Method Applied to a Singularly Perturbed Reaction-Diffusion Problem in Three Dimensions
Year: 2020
Author: Stephen Russell, Niall Madden
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 3 : pp. 297–315
Abstract
We consider a linear singularly perturbed reaction-diffusion problem in three dimensions and its numerical solution by a Galerkin finite element method with trilinear elements. The problem is discretised on a Shishkin mesh with $N$ intervals in each coordinate direction. Derivation of an error estimate for such a method is usually based on the (Shishkin) decomposition of the solution into distinct layer components. Our contribution is to provide a careful and detailed analysis of the trilinear interpolants of these components. From this analysis it is shown that, in the usual energy norm the errors converge at a rate of $\mathcal{O}$($N$−2+$ε$1/2$N$−1ln$N$). This is validated by numerical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2020-IJNAM-16860
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 3 : pp. 297–315
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Reaction-diffusion finite element Shishkin mesh three-dimensional.