**Previous:** Non-overlapping Subdomain Methods

**Up:** Domain Decomposition Methods

**Previous Page:** Non-overlapping Subdomain Methods

**Next Page:** Inexact Solves

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].