Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales
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Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales J. M. Melenk · C. Xenophontos · L. Oberbroeckling
Received: 4 January 2012 / Accepted: 3 October 2012 / Published online: 19 October 2012 © Springer Science+Business Media New York 2012
Abstract We consider a coupled system of two singularly perturbed reactiondiffusion equations, with two small parameters 0 < ε ≤ μ ≤ 1, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have boundary layers which overlap and interact, based on the relative size of ε and μ. We show how one can construct full asymptotic expansions together with error bounds that cover the complete range 0 < ε ≤ μ ≤ 1. For the present case of analytic input data, we present derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order. Keywords Singular perturbation · Multiple scales · Asymptotic expansion Mathematics Subject Classifications (2010) 34D15 · 34E05 · 65L11 · 34E13
Communicated by: M. Stynes. J. M. Melenk (B) Institut für Analysis und Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria e-mail: [email protected] C. Xenophontos Department of Mathematics and Statistics, University of Cyprus, P.O. BOX 20537, Nicosia 1678, Cyprus L. Oberbroeckling Department of Mathematics and Statistics, Loyola University Maryland, 4501 N. Charles Street, Baltimore, MD 21210, USA
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1 Introduction Singularly perturbed (SP) boundary value problems (BVPs), and their numerical approximation, have received a lot of attention in the last few decades (see, e.g., the classical texts [12, 17] on asymptotic analysis and the books [10, 11, 13], whose focus is more numerical methods for this problem class). One common feature that these problems share is the presence of boundary layers in the solution. In order for a numerical method, designed for the approximation of the solution to SP BVPs, to be considered robust it must be able to perform well, independently of the singular perturbation parameter(s). To achieve this, information about the regularity of the exact solution is utilized, and in particular, bounds on the derivatives. Such information is available in the literature for scalar SP BPVs of reaction- and convection-diffusion type in one- and two-dimensions (see, e.g., [5, 7] for scalar versions of the problem studied in the present article). For systems of SP BVPs, the bibliography is scarce, even in one-dimension; we mention here the pioneering paper [16] as well as [3, 4] and [1]—see also the relatively recent review article [2] and the references therein, for such results available to date. In all references quoted, however, only the first few derivatives of the various solution components are controlled. While this is sufficient for the analysis of methods of fixed order, it is insufficient for proving exponential convergence of a
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