An Abstract Theory for the Domain Reduction Method

Craig C. Douglas
Mathematical Sciences Department
IBM Research Division
Thomas J. Watson Research Center
P. O. Box 218
Yorktown Heights, NY 10598
and
Department of Computer Science
Yale University
P. O. Box 2158
New Haven, CT 06520
E-mail:  bells@watson.ibm.com

Jan Mandel
Computational Mathematics Group
Department of Mathematics
University of Colorado at Denver
Denver, CO  80204
E-mail:  jmandel@copper.denver.colorado.edu


Abstract:

The domain reduction method uses a finite group of symmetries of a system of
linear equations arising by discretization of partial differential equations
to obtain a decomposition into independent subproblems, which can be solved in
parallel.
This paper develops a theory for this class of methods based on known results
from group representation theory and algebras of finite groups.
It is shown that if the problem splits into subproblems based on isomorphic
subdomains, then the group of symmetries must be commutative.
General decompositions are then obtained by nesting decompositions based on
commutative groups of symmetries.