Analytic Regularity for a Singularly Perturbed Problem

This is the authors' abstract. We don't add key points for this paper.

A singularly perturbed equation of elliptic-elliptic type 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 [J. M. Melenk and C. Schwab, SIAM J. Numer. Anal., 35 (1998), pp. 1520--1557] to construct hp versions of the finite element method which feature robust exponential convergence, i.e., the rate of convergence is exponential and independent of the perturbation parameter .