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An automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady-state incompressible Navier-Stokes equations. Solutions have been obtained for the driven cavity and flow over a backward-facing step, for Reynolds numbers up to 5000 and 800, respectively. The refinement criterion is based on the local truncation error. The solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level. For driven cavity flow at Re = 1000, the adaptive refinement approach reduced the computer memory and CPU time to 20 and 40 % of the requirements of the “pure ” multigrid method. The primitive-variable formulation of the Navier-Stokes equations is used so the method can be extended easily to three dimensions. Application to other nonlinear elliptic problems is equally possible. 6 1989 Academic Press, Inc. 1