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

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SIAM Journal on Numerical Analysis


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| 21/2 + ||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 ℝ 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. © 2012 Society for Industrial and Applied Mathematics.

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