2-LEVEL PICARD AND MODIFIED PICARD METHODS FOR THE NAVIER-STOKES EQUATIONS

Iterative methods of Picard type for the Navier-Stokes equations are k nown to converge only for quite small Reynolds numbers. However, we st udy methods involving just one such iteration at general Reynolds numb ers. For the initial approximation a coarse mesh of width h(0) is used . The corrected approximation is computed by just one Picard or modifi ed Picard step on a fine mesh of width h(1). For example, h(1) may be of order O(h(0)(2)) when linear velocity elements are used. The result ing method requires the solution of a (small) system of nonlinear equa tions on the coarse mesh and only one (larger) linear system on the fi ne mesh. This two-level Picard method is proven to converge for fixed Reynolds number as h --> O. Further, the fine mesh solution satisfies a quasi-optimal error bound. (The error constants grow as Re --> infin ity, as for the usual finite element method.) One very heuristic expla nation why one step of the (divergent) Picard method might work when b eginning with a coarse mesh approximation is that the terms neglected involve lower-order derivatives; thus they are approximated with highe r accuracy on the coarse mesh. This is linked with a ''smoothing prope rty'' of the fine mesh step.