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Zbl 0828.76017
Layton, W.; Lenferink, W.
Two-level Picard and modified Picard methods for the Navier-Stokes equations. (English)
[J] Appl. Math. Comput. 69, No.2-3, 263-274 (1995). ISSN 0096-3003

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 $O(h^2_0)$ 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 $h\to 0$. Further, the fine mesh solution satisfies a quasi-optimal error bound.

MSC 2000:: *76D05 Navier-Stokes equations (fluid dynamics)
76M10 Finite element methods
65H10 Systems of nonlinear equations (numerical methods)

Keywords: finite element approximation; fine mesh solution; quasi-optimal error bound

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