##
Analysis of a Galerkin finite element method applied to a singularly perturbed reaction-diffusion problem in three dimensions

**
Stephen Russell and Niall
Madden.
**
July 2017.

### 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
$\mathcal{O}(N^{-2}+\varepsilon^{1/2}N^{-1}\ln N)$. This is validated by numerical
results.