A balanced finite element method for singularly perturbed reaction-diffusion problems

摘要

Consider the singularly perturbed linear reaction-diffusion problem -ε~ 2Δu+bu = f in Ω ? R, u = 0 on ? Ω, where d ≥ 1, the domain Ω is bounded with (when d ≥ 2) Lipschitz-continuous boundary ? Ω, and the parameter ε satisfies 0 < ε ≤ 1. It is argued that for this type of problem, the standard energy norm v & [ε~ 2|v|~ 2_ (1/2) + ||v||2/0]~ (1/2) is too weak a norm to measure adequately the errors in solutions computed by finite element methods: the multiplier ε~ 2 gives an unbalanced norm whose different components have different orders of magnitude. A balanced and stronger norm is introduced, then for d ≥ 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 ?~ 2, an error bound that is uniform in ε 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.