HP FEM for reaction-diffusion equations I: Robust exponential convergence

A singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right-hand side is analytic. We show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width O(p epsilon) is inserted near the domain boundary, that is, the rate of convergence is O(exp(-bp)) and independent of the perturbation parameter epsilon. Additionally, we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence. In particular, the spectral element method based on the use of a Gauss-Lobatto quadrature rule with (p + 1) x (p + 1) points yields robust exponential convergence.