Weighted max norms, splittings, and overlapping additive Schwarz iterations

Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterat ions with overlapping blocks for the solution of Ax = b, when the coefficie nt matrix A is an M-matrix. The case of inexact local solvers is also cover ed. These bounds are analogous to those that exist using A-norms when the m atrix A is symmetric positive definite. A new theorem concerning P-regular splittings is presented which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Ad ditive Schwarz Iterations. This representation makes a connection with mult isplitting methods. With this representation, and using a comparison theore m, it is shown that a coarse grid correction improves the convergence of Ad ditive Schwarz Iterations when measured in weighted max norm.