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 - <p style="text-align: justify;">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 $\mathcal{O}$($N$<sup>−2</sup>+$ε$<sup>1/2</sup>$N$<sup>−1</sup>ln$N$). This is validated by numerical results.</p>