Volume 17, Issue 3
Analysis of a Galerkin Finite Element Method Applied to a Singularly Perturbed Reaction-Diffusion Problem in Three Dimensions

Int. J. Numer. Anal. Mod., 17 (2020), pp. 297-315.

Published online: 2020-05

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• 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.

• Keywords

Reaction-diffusion, finite element, Shishkin mesh, three-dimensional.

65N12, 65N15, 65N30

russellstephen17@gmail.com (Stephen Russell)

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@Article{IJNAM-17-297, author = {Stephen and Russell and russellstephen17@gmail.com and 7572 and Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Haidian District, Beijing 100084, China and Stephen Russell and Niall and Madden and Niall.Madden@NUIGalway.ie and 7573 and School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland and Niall Madden}, title = {Analysis of a Galerkin Finite Element Method Applied to a Singularly Perturbed Reaction-Diffusion Problem in Three Dimensions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {3}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/16860.html} }
TY - JOUR T1 - Analysis of a Galerkin Finite Element Method Applied to a Singularly Perturbed Reaction-Diffusion Problem in Three Dimensions AU - Russell , Stephen AU - Madden , Niall JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 297 EP - 315 PY - 2020 DA - 2020/05 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/16860.html KW - Reaction-diffusion, finite element, Shishkin mesh, three-dimensional. AB -

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.

Stephen Russell & Niall Madden . (2020). Analysis of a Galerkin Finite Element Method Applied to a Singularly Perturbed Reaction-Diffusion Problem in Three Dimensions. International Journal of Numerical Analysis and Modeling. 17 (3). 297-315. doi:
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