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An automatic adaptive refinement technique has been combined with the multigrid method and applied to the steady-state Navier-Stokes equations. The approach is both very efficient and stable. The adaptive refinement criterion is based on the local truncation error. For two-dimensional driven cavity flow at Re = 1000, the technique reduced the computer memory and CPU-time to 20 and 40% of the requirements of the 'pure' multigrid method. Second-order solutions have been obtained for Reynolds numbers up to 5000. Application to other nonlinear elliptic problems is equally possible.

One dimensional tests indicate that the application of one step of the cyclic reduction method to central-difference approximations of convection-diffusion equations effectively transforms a highly convective problem into a diffusive one. The resulting system of equations can then be solved efficiently by the classical SOR technique. The approach can be extended to two dimensions; however, care must be taken to ensure that the iterative system has Young's Property (A). This involves basing the SOR sweeping on updating two lines at a time. The application to some two-dimensional test problems shows that this approach has very good convergence properties compared with other relaxation methods. Finally, the two-dimensional driven cavity flow problem is treated. Converged solutions are obtained for Reynolds numbers up to 10000.