2-LEVEL PICARD AND MODIFIED PICARD METHODS FOR THE NAVIER-STOKES EQUATIONS
2-LEVEL PICARD AND MODIFIED PICARD METHODS FOR THE NAVIER-STOKES EQUATIONS
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DOI：
10.1016/0096-3003(94)00134-p
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发表时间：
1995-05-01
影响因子：
4
通讯作者：
LENFERINK, W
中科院分区：
文献类型：
--
作者：
LAYTON, W;LENFERINK, W
已知求解Navier-Stokes方程的Picard型迭代法仅在雷诺数很小时才收敛。然而，我们研究的方法只涉及一个这样的迭代在一般雷诺数。对于初始近似，使用宽度为h（0）的粗网格。校正近似值仅通过宽度为h（1）的精细网格上的一个Picard或修改的Picard步长来计算。例如，当使用线速度元件时，h（1）可以是O（h（0）（2））阶。该方法在粗网格上只需要求解一个非线性方程组，而在细网格上只需要求解一个线性方程组，当雷诺数h> 0时，该方法是收敛的。此外，细网格解决方案满足准最优误差界。(The误差常数随着Re -->无穷大而增长，与通常的有限元法一样。）一个非常启发性的解释是，为什么（发散）Picard方法的一个步骤可能在粗网格近似开始时起作用，因为忽略的项涉及低阶导数;因此它们在粗网格上以更高的精度近似。这与精细网格步骤的“平滑属性”相关联。
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 --> O. Further, the fine mesh solution satisfies a quasi-optimal error bound. (The error constants grow as Re --> infinity, 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.