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.