Robust exponential convergence of hp FEM for singularly perturbed reaction-diffusion systems with multiple scales

Abstract

We consider a coupled system of two singularly perturbed reaction-diffusion equations in one dimension. Associated with the two singular perturbation parameters 0 < ε ≤ μ ≤ 1 are boundary layers of length scales O(ε) and O(μ). We propose and analyse an hp finite element scheme which includes elements of size O(ε p) and O(μ p) near the boundary, where p is the degree of the approximating polynomials. We show that under the assumption of analytic input data, the method yields exponential rates of convergence, independently of ε and μ and independently of the relative size of ε to μ. In particular, the full range 0 < ε ≤ μ ≤ 1 is covered by our analysis. Numerical computations supporting the theory are also presented. © 2012 The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.