Summary.
Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M-matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix 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 Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm.
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Received March 13, 1998 / Revised version received January 26, 1999
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Frommer, A., Szyld, D. Weighted max norms, splittings, and overlapping additive Schwarz iterations. Numer. Math. 83, 259–278 (1999). https://doi.org/10.1007/s002110050449
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DOI: https://doi.org/10.1007/s002110050449