A Balanced Finite Element Method for Singularly Perturbed Reaction-Diffusion Problems

Consider the singularly perturbed linear reaction-diffusion problem $-\varepsilon^2 \Delta u + bu = f$ in $\Omega \subset \mathbb{R}^d$, $u=0$ on $\partial\Omega$, where $d \ge 1$, the domain $\Omega$ is bounded with (when $d \ge 2$) Lipschitz-continuous boundary $\partial\Omega$, and the parameter $\varepsilon$ satisfies $0 < \varepsilon \ll 1$. It is argued that for this type of problem, the standard energy norm $v \mapsto[\varepsilon^2|v|_1^2 + \|v\|_0^2]^{1/2}$ is too weak a norm to measure adequately the errors in solutions computed by finite element methods: the multiplier $\varepsilon^2$ gives an unbalanced norm whose different components have different orders of magnitude. A balanced and stronger norm is introduced, then for $d \ge 2$ a mixed finite element method is constructed whose solution is quasi-optimal in this new norm. For a problem posed on the unit square in $\mathbb{R}^2$, an error bound that is uniform in $\varepsilon$ is proved when the new method is implemented on a Shishkin mesh. Numerical results are presented to show the superiority of the new method over the standard mixed finite element method on the same mesh for this singularly perturbed problem.