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\bf A DECOMPOSITION METHOD FOR SINGULARLY PERTURBED
REACTION-DIFFUSION EQUATIONS
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Irina V. Tselishcheva, ~~ Grigorii I. Shishkin\\[1ex]
{\it Institute of Mathematics and Mechanics,
Russian Academy of Sciences, Ekaterinburg 620219,  Russia}
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A Dirichlet problem for singularly perturbed elliptic equations
is considered on a rectangle. The parameter $\eps$ multiplying
the highest derivatives takes any values in
the half-interval (0,1]. The solution of such reaction-diffusion problem
exhibits boundary layers as $\eps\to 0$.  If we apply any discretization
technique to this $\eps$-dependent problem, then the behaviour of
the discretization depends on the parameter $\eps$. In particular,
classical difference approximations on uniform meshes lead to
discretizations that are worthless because of large errors
if the perturbation parameter $\eps$ is close to some critical value.
We are interested in robust and efficient methods that work for all
values of the parameter, no matter how small, and
yield error bounds, in the maximum norm,
which are independent of $\eps$. Such techniques include both
suitable discrete approximations for the boundary value
problems and efficient numerical methods for solving the discrete
problems which arise from the approximation.

In the present paper we propose and study one approach how to
construct $\eps$-uniform discrete approximations for the problem
under consideration. This approach is based on the splitting of the
singularities and on the domain decomposition technique.
As is known, the solution of the problem can be represented as a sum of its
smooth and singular components.
Moreover, the latter of these components, as the boundary layer function,
is representable in the form of a sum of sufficiently simple
functions which contain regular and corner boundary layers.
Such decomposition of the solution allows us to reduce the original
problem to an equivalent system of auxiliary Dirichlet's problems,
which are solved sequentially one by one, and whose solutions have a
sufficiently simple behaviour.  Because the regular and corner layers
differ essentially from zero only in the neighbourhood of the sides
and vertices of the rectangle, and, in addition, they decreases
exponentially as the point $(x_1,x_2)$ moves away from the corresponding
sets, then we can consider the relevant auxiliary problems only on
sufficiently small (rectangle) subdomains, but not on the whole
domain of definition. Such decomposition of the domain permits to
simplify the numerical solution of the resulting system and, therefore,
of the original problem.

In a proper way, we determine the subdomains on which the auxiliary
problems are solved.  To approximate these problems, we use classical
difference approximations on uniform meshes. These schemes are easy
and convenient for implementation. We give the conditions under
which the discrete solutions, constructed by this way, converge
$\eps$-uniformly to the solution of the boundary value problem.\\

This work was supported in part by the Russian Foundation for
Basic Research under Grant N 98-01-00362.
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