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

Consider the singularly perturbed linear reaction-diffusion problem −ε2Δu+ bu = f in Ω ⊂ Rd, u = 0 on ∂Ω, where d ≥ 1, the domain Ω is bounded with (when d ≥ 2) Lipschitzcontinuous 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|v|1+‖v‖0] is too weak a norm to measure adequately the errors in solutions computed by finite element methods: the multiplier ε2 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 quasioptimal in this new norm. For a problem posed on the unit square in R2, 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.