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\title{Parallel domain decomposition approach to 3D boundary value problems 
and implementations on parallel computers}
\author{Krassimir Georgiev \thanks 
{The author was supported in part by NATO Scientific Affairs Division 
and the Bulgarian Ministry of Education and Science under Grant I-505/95}}
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\centerline{A B S T R A C T}
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In recent years there has been a considerable development in 
Domain Decomposition (DD) methods for
nume\-rical approximation of partial differential equations in the framework of
finite differences and finite elements methods. Among the reasons
underly the interest in these techniques we mention the possibilities  of
determining effective pa\-rallel algorithms and implementation on modern
high-speed
computers mainly parallel computers with shared and distributed memory 
including here clusters of workstations.

In this paper we are mainly concerned with the additive overlapping domain 
decomposition algorithms for the numerical solution of 3D elasticity systems
and their implementation on parallel computers 
with shared and distributed memory including here clusters of workstations.
Nevertheless, all the ideas into the 
algorithms and their implementation are valid for many others three dimensional 
elliptic boundary value problems with strongly varying coefficients.

The system of linear equations has to be solved after a finite element 
method discretization is symmetric and positive definite and the Preconditioned 
Conjugate Gradient method with a preconditioner constructed via Domain 
Decomposition is used. The algorithm is highly parallelizable.
The Message Passing Interface (MPI) standard both for shared and
distributed memory parallel computers is used.

The numerical experiments reported in this paper are obtained using
two symmetric multiprocessor systems: SUN Enterprise 3000 and SUN 
SPARCstation 20 
as well as the parallel computer with distributed memory IBM SP Power 2. 
A set of tables and
figures illustrate the performed parallel numerical tests. They well
demonstrate the parallel properties of the proposed 
algorithm as well as the robustness of the developed codes.

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