The hp finite element approximation of a weakly coupled system of two singularly perturbed reaction-diffusion equations

We consider the approximation of a weakly coupled system of two singularly perturbed reaction-diffusion equations, with the finite element method. The first differential equation contains a small parameter multiplying the highest derivative, while the second does not. As a result, the first component of the solution to the system will contain boundary layers and our goal is to construct and analyze an hp finite element scheme which converges uniformly with respect to the singular perturbation parameter. In particular, our scheme 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, as p → ∞. Numerical computations supporting the theory are also presented.