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Multisplitting Methods: Optimal Schemes for the Unknowns in a Given Overlap

Consider a linear algebraic problem where the set of unknowns is a union of subsets. Let the coefficient matrix have a splitting associated with each subset. The traditional multisplitting method forms a weighted sum, over the overlapping unknowns, of the iterates for each such splitting to obtain a single parallel iterative method. An optimal alternative to the weighted sums will be presented. Convergence of this new form of multisplitting (MS) method can be studied for both symmetric positive definite (SPD) matrices and M-matrices. Applications to PDEs and the equilibrium equations for fluid flow in a driven cavity will be presented.