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Robust high-order accurate numerical methods for singularly perturbed problems

Шишкин Г.И.

Институт математики и механики УрО РАН (Екатеринбург)

documentstyle[12pt]{article} itle{Robust high-order accurate numerical methods for singularly perturbed problems} author{Grigory I. Shishkin} date{} egin{document} maketitle

Solutions of singularly perturbed convection-diffusion problems have boundary layers controlled by a perturbation parameter $varepsilon$. This parameter multiplies the highest space derivatives and is related to the thickness of the boundary layer. It is well known that standard numerical methods give errors in the solutions which grow and become comparable with the exact solution when $varepsilon$ becomes small. That is why robust numerical methods, i.e., methods for which errors are independent of the parameter $varepsilon$ (or in short, $varepsilon$-uniform methods), are very important. At present robust numerical methods have been developed for various problems with boundary layers. However, the order of accuracy for such $varepsilon$-uniform special numerical methods is too low and does not exceed one, which is a main restriction to use such methods. Thus, the development of $varepsilon$-uniformly convergent methods with the accuracy order higher than unity is an important task.

To construct high-order accurate $varepsilon$-uniform numerical methods we use three approaches: (a) the defect-correction method, (b) the Richardson extrapolation method, (c) a method based on the asymptotic expansion technique. When constructing special schemes, we use piecewise uniform meshes condensing in a neighbourhood of the boundary layers. We discuss the base principles underlying such approaches, and also expose some results derived on those principles. The approaches (a) and (b) are well elaborated, and the new approach (c) is under development. In the case (c) for small values of the parameter, we use approximations to ``auxiliary'' subproblems which describe the main terms of asymptotic representations of the solution in a neighbourhood of the boundary layer and outside it.

Acknowledgements. This research was supported in part by the Russian Foundation for Basic Research under grant 04-01-00578 and by the Dutch Research Organisation NWO under grant No. 047.016.008. end{document}

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