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

Abstract

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 ε-weighted H1-norm, where the weighting in terms of the singular perturbation parameter ε is such that, in contrast to the standard energy norm, boundary layer contributions do not vanish in the limit ε → 0. Robust exponential convergence in the maximum norm is also established. We illustrate the theoretical findings with two numerical experiments. © 2015, Springer-Verlag Italia.