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 < epsilon less than or equal to mu less than or equal to 1 are boundary layers of length scales O( epsilon ) and O( mu ). We propose and analyse an hp finite element scheme which includes elements of size O( epsilon p) and O( mu 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 epsilon and mu and independently of the relative size of epsilon to mu . In particular, the full range 0 < epsilon less than or equal to mu less than or equal to 1 is covered by our analysis. Numerical computations supporting the theory are also presented.