Balanced-Norm and Energy-Norm Error Analyses for a Backward Euler/FEM Solving a Singularly Perturbed Parabolic Reaction-Diffusion Problem

Abstract

In the derivation of error bounds, uniformly in the singular perturbation parameter $\varepsilon$, for finite element methods (FEMs) applied to elliptic singularly perturbed linear reaction-diffusion problems, the usual energy norm is unsatisfactory since it is essentially no stronger than the $L^2$ norm. Consequently various researchers have analysed errors in FEM solutions, uniformly in $\varepsilon$, using balanced norms whose $H^1$ component is weighted correctly to maintain its influence. But the derivation of energy and balanced-norm error bounds for FEM solutions of singularly perturbed reaction-diffusion problems is confined almost entirely to steady-state elliptic problems — little has been proved for time-dependent parabolic singularly perturbed problems. The present paper addresses this gap in the literature: the backward Euler method in time, combined with a bilinear FEM on a spatial Shishkin mesh, is applied to solve a parabolic singularly perturbed reaction-diffusion problem, and energy-norm and balanced-norm error estimates, which are uniform in the singular perturbation parameter $\varepsilon$, are derived — these results are stronger than any previous results of the same type. Furthermore, numerical experiments demonstrate the sharpness of our error bounds.