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An hp finite element method for singularly perturbed systems of reactiondiffusion equations
Author(s) -
Oberbroeckling Lisa,
Xenophontos Christos
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700348
Subject(s) - singular perturbation , mathematics , finite element method , exponential function , perturbation (astronomy) , mathematical analysis , reaction–diffusion system , rate of convergence , boundary value problem , boundary knot method , convergence (economics) , boundary (topology) , boundary element method , physics , computer science , computer network , channel (broadcasting) , quantum mechanics , economics , thermodynamics , economic growth
We consider the approximation of a coupled system of two singularly perturbed reaction‐diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order hp finite element scheme which includes elements of size O ( εp ) and O ( μp ) near the boundary, where ε , μ are the singular perturbation parameters and p is the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p → ∞, independently of ε and μ . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)