Analytic regularity for a singularly perturbed problem

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 th e right-hand side is analytic. We give asymptotic expansions of the solutio n and new error bounds that are uniform in the perturbation parameter as we ll as in the expansion order. Additionally, we provide growth estimates for higher derivatives of the solution where the dependence on the perturbatio n 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 feat ure robust exponential convergence, i.e., the rate of convergence is expone ntial and independent of the perturbation parameter epsilon.