定常Navier-Stokes方程的三种两层稳定有限元算法计算效率分析

COMPUTATION EFFICIENCY ON THREE KINDS OF TWO-LEVEL STABILIZED FINITE ELEMENT METHODS FOR STATIONARY NAVIER-STOKES EQUATIONS

讨论分析了定常Navier-Stokes(N-S)方程的三种两层稳定有限元算法.它们将局部高斯积分稳定化技术和两层算法的思想充分结合,采用不满足Inf-Sup条件的低次等价有限元P_1-P_1或Q_1-Q_1对N-S方程进行数值求解,在粗网格上解定常N-S方程,在细网格上只需求解一个Stokes方程.误差分析和数值实验都表明,当它们的粗、细网格尺度比分别为H=h^(1/3)| logh|^(-1/6),H=O(h^(1/2))和H=O(h^(1/2))时,它们与在细网格上的标准有限元算法具有相同的收敛速度.而两层稳定有限元算法却节省了大量的计算时间.相比之下,简单两层稳定有限元算法具有更高的计算效率,Oseen两层算法次之,Newton两层算法较低而且进一步发现较小粘性系数对Newton两层算法数值精度影响较大.

In this paper, three kinds of two-level stabilized finite element methods based on local Gauss integral technique for the two-dimensional stationary Navier-Stokes equations approx- imated by the lowest equal-order P1-P1 or Q1-Q1 elements while do not satisfy the inf-sup condion are considered. The two-level methods consist of solving a small non-linear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths H = hl/3｜ log h｜-1/6, H = O（h1/2） and H = O（h1/2） Therefore, the two-level methods are of practical importance in scientific computation. Finally, the performance of three kinds of two-level stabilized methods are compared in efficiency and precision aspects by a seriesof numerical experiments. The conclusion is that the simple two-level stabilized methods is best than the others in accuracy and efficiency. And, there is poor numerical accuracy for the Newton algorithm to N-S equations with low viscosity coefficient.