An Abstract Theory for the Domain Reduction Method

Craig C. Douglas
IBM T. J. Watson Research Center
Yorktown Heights, NY, USA and
Computer Science Department
Yale University
New Haven, CT, USA

Jan Mandel
Computational Mathematics Group
Department of Mathematics
University of Colorado at Denver
Denver, CO 80204


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.

Contributed October 21, 1991. Updated April 4, 1992.