Two-level Picard and modified Picard methods for the Navier-Stokes equations

Abstract

Iterative methods of Picard type for the Navier-Stokes equations are known to converge only for quite small Reynolds numbers. However, we study methods involving just one such iteration at general Reynolds numbers. For the initial approximation a coarse mesh of width h 0 is used. The corrected approximation is computed by just one Picard or modified 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 resulting method requires the solution of a (small) system of nonlinear equations on the coarse mesh and only one (larger) linear system on the fine mesh. This two-level Picard method is proven to converge for fixed Reynolds number as h → 0. Further, the fine mesh solution satisfies a quasi-optimal error bound. (The error constants grow as Re → ∞, as for the usual finite element method.) One very heuristic explanation why one step of the (divergent) Picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower-order derivatives; thus they are approximated with higher accuracy on the coarse mesh. This is linked with a “smoothing property” of the fine mesh step.