An abstract theory for the domain reduction method

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. The main theoretical result is 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.