This lecture presents domain decomposition iterative methods for partial differential equations --- algorithms designed to bridge the gap between physical applications involving, at some fundamental level, all-to-all data dependencies between the unknown quantities and distributed computer architectures with high communication-to-computation cost ratios. Century-old threads of pre-computational mathematical progress are woven with the fabric of contemporary computer architecture and the resulting ``Newton-Krylov-Schwarz'' methods are illustrated on problems from computational fluid dynamics.

The one-hour lecture requires an overhead slide projector. An optional 10-minute audience exercise in the convergence rate of iterative methods for the diffusion equation requires a blackboard over which eight people can be spread out.