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Численное решение дифференциальных и интегральных уравнений

Domain decomposition methods for quasilinear singularly perturbed parabolic equations of reaction-diffusion type in a composed domain

Целищева И.В.

Институт математики и механики УрО РАН (Екатеринбург)

documentstyle[12pt]{article} title{Domain decomposition methods for quasilinear singularly perturbed parabolic equations of reaction-diffusion type in a composed domain} author{Pieter W. Hemker, Grigory I. Shishkin, Irina V. Tselishcheva} date{} begin{document} maketitle

In a space-time domain composed of two rectangles, we consider a singularly perturbed boundary value problem for a quasilinear parabolic reaction-diffusion equation. The highest space derivative is multiplied by the perturbation parameter $varepsilon$ taking arbitrary values in the half-open interval (0,1]. The conjugation conditions reflecting the continuity of the solution and of the diffusion flux are given on the interface between the subdomains. As $varepsilon ightarrow 0$, the solution exhibits boundary layers (in a neighbourhood of the parabolic boundary) and a transient (interior) layer, which occurs on both sides of the interface boundary. Because of the thin layers, standard finite difference methods applied to problems of this class yield unsatisfactorily large errors for small $varepsilon$. We construct a special (nonlinear) difference scheme that converges $varepsilon$-uniformly at the rate $O(N^{-2}ln^2 N + N_0^{-1})$, where $N$ and $N_0$ are the numbers of mesh intervals in the space and time meshes, respectively. For this we use standard difference approximations on piecewise uniform meshes condensing in the layer regions. Based on this scheme, we construct domain decomposition (DD) schemes, in which the intermediate (linearized) problems on the overlapping subdomains can be solved sequentially and in parallel, independently of each other. We give conditions under which the overlapping Schwarz method is robust in the sense that its solutions converge $varepsilon$-uniformly as the number of mesh points and the number of iterations grow, and the number of iterations required for convergence is independent of $varepsilon$. The iterative DD schemes inherit the property of $varepsilon$-uniform convergence of the same order.

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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