One of the successes of domain decomposition theory has been
in the construction of coarse
global components of the preconditioner, which allow for an iteration
count that is independent
of the number of subdomains as well as growing quite slowly with the
number of degrees of
freedom in individual subdomains. It has also been possible to
establish bounds which are
independent of jumps in the material parameters across the interface
formed by the boundaries
of the subdomains.

We face new issues when there is a pair of material parameters as in
elasticity and in problems
posed in H(div) and H(curl). We will survey some new results and
algorithmic ideas for these
three cases.