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Abstract:
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In the theory for domain decomposition methods, we have
previously often assumed that each subdomain is the
union of a small set of coarse
shape-regular triangles or tetrahedra. In this talk, we discuss
recent progress which makes it possible to analyze cases with
irregular subdomains such as those provided by mesh partitioners.
Our goal is to extend our analytic tools to problems on
subdomains that might not even be Lipschitz and to characterize the
rates
of convergence of our methods in terms of a few, easy to
understand, geometric parameters of the subregions. For two dimensions,
we have already obtained some best possible results for scalar elliptic
and linear elasticity problems: the subdomains should be
John or Jones domains and the rate of
convergence is determined using the parameters that define
such domains and that of an isoperametric inequality. Progress
on three dimensions will also be reported.
New results have also recently been obtained concerning variants of
classical two-level additive Schwarz preconditioners.
Our family of overlapping Schwarz methods,
borrows and extends coarse spaces from older iterative substructuring
methods, i.e., methods based on non-overlapping subdomains.
The local components of these preconditioners, on the other hand,
are based on Dirichlet problems defined on a set of
overlapping subdomains which cover the original domain.
Our methods are robust even in the presence of large changes, between
subdomains, of the materials being modeled in the finite element models.
An extra attraction is that our methods can be applied
directly to problems where the stiffness matrix is available
only in its fully assembled form.
These new methods are being used successfully
as part of a production-level iterative solver in the parallel
structural
dynamics code Salinas, developed at Sandia
National Laboratories in Albuquerque, NM.
Several applications of the new tools will also be discussed.
They include new results on
almost incompressible elasticity and mixed finite elements
using spaces of discontinuous pressures and
Maxwell's equations in two dimensions using edge elements.
Our work is carried out in close collaboration with
Clark R. Dohrmann of the Sandia National Laboratories,
Albuquerque, NM and
Axel Klawonn and Oliver Rheinbach
of the University of Duisburg-Essen, Germany.
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