@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} }