Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations

The hp-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed on an interval or a two-dimensional domain with an analytic boundary. On suitably designed Spectral Boundary Layer meshes, robust exponential convergence in a "balanced" norm is shown. This "balanced" norm is an epsilon-weighted H-1-norm, where the weighting in terms of the singular perturbation parameter epsilon is such that, in contrast to the standard energy norm, boundary layer contributions do not vanish in the limit epsilon -> 0. Robust exponential convergence in the maximum norm is also established. We illustrate the theoretical findings with two numerical experiments.