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Multiplicative Schwarz Methods

As mentioned before, the Additive Schwarz preconditioner can be viewed as an overlapping block Jacobi preconditioner. Analogously, one can define a multiplicative Schwarz preconditioner which corresponds to a symmetric block Gauss-Seidel version. That is, the solves on each subdomain are performed sequentially, using the most current iterates as boundary conditions from neighboring subdomains. Even without conjugate gradient acceleration, the multiplicative method can take many fewer iterations than the additive version. However, the multiplicative version is not as parallelizable, although the degree of parallelism can be increased by trading off the convergence rate through multi-coloring the subdomains. The theory can be found in Bramble, et al. [36].