On the hp finite element approximation of systems of singularly perturbed reaction-diffusion equations

We consider the approximation of systems of singularly perturbed reaction-diffusion equations, with the finite element method. The solution to such problems contains, in general, boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. In this article, we focus on the case when the singular perturbation parameters are equal and adapt the analysis of the corresponding scalar problem from [9], to construct an hp finite element scheme which includes elements of size O(εp) near the boundary, where ε is the singular perturbation parameter and 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. Numerical computations supporting the theory are also presented.