The Domain Reduction Method: High Way Reduction in Three Dimensions and Convergence with Inexact Solvers

We study a method for parallel solution of elliptic partial di erential equations which decomposes the problem into a number of independent subproblems on subspaces of the underlying solution space. Using symmetries of the domain, we obtain up to 64 such subproblems for a 3 dimensional cube and the method reduces to a direct solver. In the general case, or when the subproblems are solved only approximately, the method becomes an iterative method or can be used as a preconditioner. Bounds on the resulting convergence factors and condition numbers are given.