The paper considers standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered (Cahouet&Chabard and augmented Lagrangian), and one investigates whether these methods can compete with traditional pressure-correction and velocity-correction methods in terms of CPU time per degree of freedom and per time step. Numerical tests on fine unstructured meshes (68 millions degrees of freedoms) demonstrate convergence rates that are independent of the mesh size and improve with the Reynolds number. Three conclusions are drawn from the paper: (1) Although very good parallel scalability is observed for the augmented Lagrangian method, thorough tests on large problems reveal that the overall CPU time per degree of freedom and per time step is best for the standard Cahouet&Chabar preconditioner. (2) Whether solving the pressure Schur complement problem or solving the full couple system at once does not make any significant difference in term of CPU time per degree of freedom and per time step. (3) All the methods tested in the paper, whether matrix-free or not, are on average 30 times slower than traditional pressure-correction and velocity-correction methods. Hence, although all these methods are very efficient for solving steady state problems, they are not yet competitive for solving time-dependent problems.

翻译：本文研究了求解时变不可压缩Navier-Stokes方程时空离散化所产生的广义Stokes问题的标准迭代方法。探讨了多种预处理技术（Cahouet&Chabard预处理及增广拉格朗日法），并系统评估了这些方法在单位自由度单位时间步长CPU耗时方面能否与传统压力修正法和速度修正法竞争。在精细非结构网格（6800万自由度）上的数值实验表明，所提方法的收敛速率与网格尺寸无关且随雷诺数提升而改善。本文得出三项结论：（1）虽然增广拉格朗日法展现出优异的并行可扩展性，但大规模问题测试表明单位自由度单位时间步长总CPU耗时最优者仍为标准Cahouet&Chabard预处理器；（2）无论是求解压力Schur补问题还是直接求解完整耦合系统，在单位自由度单位时间步长CPU耗时方面均无显著差异；（3）本文测试的所有方法（无论是否采用矩阵自由格式）平均比传统压力修正法和速度修正法慢30倍。因此，尽管这些方法在求解稳态问题时效率卓越，但对于时变问题求解尚不具备竞争力。