A BALANCED FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS

Consider the singularly perturbed linear reaction-diffusion problem -epsilon(2)Delta u+bu - f in Omega subset of R-d, u = 0 on partial derivative Omega, where d >= 1, the domain Omega is bounded with (when d >= 2) Lipschitz-continuous boundary partial derivative Omega, and the parameter epsilon satisfies 0 < epsilon << 1. It is argued that for this type of problem, the standard energy norm nu bar right arrow [epsilon(2)vertical bar v vertical bar(2)(1) + parallel to nu parallel to(2)(0)](1/2) is too weak a norm to measure adequately the errors in solutions computed by finite element methods: the multiplier epsilon(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 e 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.