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J. Melenk, C. Xenophontos, L. Oberbroeckling:
"Robust exponential convergence of hp-FEM for singularly perturbed reaction diffusion systems with multiple scales";
in: "ASC Report 31/2011", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

We consider a coupled system of two singularly perturbed reaction-diffusion equations in one dimension. Associated with the two singular perturbation parameters $0 < \varepsilon \leq \mu \leq 1$, are boundary layers of length scales $O(\varepsilon)$ and $O(\mu)$.
We propose and analyze an $hp$ finite
element scheme which includes elements of size $O(\e p)$ and $O(\mu p)$ near the boundary, where $p$ is the degree of the approximating polynomials.
We show that under the
assumption of analytic input data, the method yields \emph{exponential} rates of convergence, independently of $\varepsilon$ and $\mu$ and independently of the relative size of $\varepsilon$ to $\mu$.
In particular, the full range $0 < \varepsilon \leq \mu \leq 1$ is covered by our analysis.
Numerical computations supporting the theory are also presented.

http://www.asc.tuwien.ac.at/preprint/2011/asc31x2011.pdf