Atlas Mathematical Conference Abstracts - Document caen-03 | Copyright © 2000 by Irina Tselishcheva and Grigorii I. Shishkin (Institute of Mathematics and Mechanics, Ural Branch of RAS, Ekaterinburg, Russia; email:
Second Conference on Numerical Analysis and Applications
June 11 - 15, 2000
University of Rousse
Rousee, Bulgaria
Conference Organizers
Plamen Yalamov, Marcin Paprzycki and Lubin Vulkov

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Domain decomposition finite difference method for singularly perturbed elliptic equations in composed domains
presented by
Irina Tselishcheva
Institute of Mathematics and Mechanics, Ural Branch of RAS, Ekaterinburg 620219, Russia; email:
joint research with
Grigorii I. Shishkin (Institute of Mathematics and Mechanics, Ural Branch of RAS, Ekaterinburg, Russia; email:

Numerical modelling and studying of stationary heat and mass transfer processes in composite materials often bring us to singularly perturbed problems in composed domains, that is, to elliptic equations with discontinuous coefficients and a small parameter e multiplying the highest derivatives. For such problems the application of numerical methods based on a domain decomposition (DD) technique seems quite reasonable; the original domain is naturally decomposed into several subdomains with smooth coefficients. Due to the presence of transition and boundary layers, standard numerical methods yield large errors. By this reason, we need for special numerical methods with errors independent of the parameter e, i.e., methods convergent e-uniformly. For the above problems considered on sufficiently simple canonical regions, we construct domain decomposition finite difference schemes that converge e-uniformly. To this end, we use classical finite difference approximations on piecewise uniform grids, which are a priori refined in the transition and boundary layers. We study parameters of the DD schemes for which the number of iterations in the iterative numerical process is independent of the perturbation parameter.

The work has been supported by the Russian Foundation for Basic Research under Grant N 98-01-00362.

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Date received: February 4, 2000

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