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

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 Λ(ε) and Λ(μ). We propose and analyse an hp finite element scheme which includes elements of size Λ(εp) and Λ(μp) 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.