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Viscous fluid flow is often smooth in most of the domain, with regions of rapid variation confined to some rather narrow zones in the field. These zones (boundary layers, shocks, etc.) cause problems during numerical solution of the equations governing the flow. The patched adaptive mesh refinement technique, devised at Stanford by Oliger, et al., copes with these sources of error efficiently by refining the computational grid locally. This is done by creating separate fine grids for every region of large error. Because of the success of this approach, a project was started to extend its applicability to geometrically complex domains. As patched adaptive mesh refinement already entails multiple grids, it was decided that geometrical complexity would also be tacked using several grids. An arbitrarily shaped domain typically cannot be covered by a single grid without severe distortion, but a covering can be established with only mildly curved grids if more than one grid is allowed. Communication between these grids then becomes an issue, as well as their creation. In this project various types of communications between grids, based on the Schwarz Alternating Procedure (SWAP), are examined for solving steady, two-dimensional incompressible-flow problems. Of these, the traditional SWAP on sets of overlapping grids works best and gives accurate results, despite the use of nonconservative interpolation procedures between grids. When reentrant problems occur, the pressures between grids may not match. A pressure- communication scheme is devised which solves this difficulty.