Abstract: This paper presents some preliminary work on the application of Schwarz alternating method, combined with a multigrid solver, for solving the steady 2D viscous incompressible Navier-Stokes equations. The difficulty caused by the violation of the compatibility condition imposed on the normal velocity at the boundary, which arises when subdomain problems are not solved exactly at each iteration, is surrounded by adding artificial compressiblity constants to the continuity equation each time when a subdomain problem is to be solved. These constants permit to maintain the compatibility condition of the subdomain problems and one can easily show that they tend to zero when the iteration number tends to infinity (in fact, they can be expressed explicitly as functions of the iteration number and some geometrical parameters of the computational domain). Numerical experiences have been made for flows in a channel with a facing step.

Key words: iteration, continuity, Schwarz, infinity, difficulty, exactly, maintain, adding, incompressible, constants