Some Schwarz Methods for Symmetric And Nonsymmetric Elliptic Problems
Some Schwarz Methods for Symmetric And Nonsymmetric Elliptic Problems
Olof B Widlund
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' Courant Institute of Mathematical Science, New York University, 251 Mercer Street, New York, N. Y. 10012. This work was supported in part by the National Science Foundation under Grant NSF- CCR-8903003 and in part by the U. S. Department of Energy under contract DE-FG02-88ER250553 at the Courant Mathematics and Computing Laboratory. 1 This paper is organized as follows. In Section 2, we first briefly review the theory for the positive definite, symmetric case and then, to provide an example, ...turn to a special case already discussed in our first papers on Schwarz methods, cf. Dryja and Widlund [15], [16]. This additive method uses two levels of discretization corre- sponding to a coarse space and a number of local finite element spaces. The former plays a role similar to the coarse mesh solver in a multigrid method; it provides global transportation of information across the region. In our original proof that the condition number for this Schwarz method is inde- pendent of the number of subspaces and the mesh size, it is important to assume that there is a relatively generous overlap between the subregions.
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