Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion equations
Provides a theoretical guarantee for robust exponential convergence in a stronger norm, addressing a known bottleneck in numerical methods for singularly perturbed problems.
The paper proves robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations, using Spectral Boundary Layer meshes. Numerical experiments confirm the theory.
The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments.