hp FEM for Reaction-Diffusion Equations. II: Regularity Theory

Abstract

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, the boundary data are analytic, and the right hand side is analytic. We give asymptotic expansions of the solution and new error bounds that are uniform in the perturbation parameter as well as in the expansion order. Additionally, we provide growth estimates for higher derivatives of the solution where the dependence on the perturbation parameter appears explicitly. These error bounds and growth estimates are used in the first part of this work to construct hp versions of the finite element method which feature {\em robust exponential convergence}, i.e., the rate of convergence is exponential and independent of the perturbation parameter $\varepsilon$.