The paper compares 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 (Schur complement, fully coupled system, with and without augmented Lagrangian). One investigates whether these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (number of degrees of freedom per time step per core per second). Numerical tests on fine unstructured meshes (68 millions degrees of freedoms) demonstrate GMRES/CG convergence rates that are independent of the mesh size and improve with the Reynolds number for most methods. Three conclusions are drawn: (1) Whether solving the pressure Schur complement or the fully coupled system does not make any significant difference in terms of throughput. (2) Although very good parallel scalability is observed for the augmented Lagrangian method, the best throughput is achieved without using the augmented Lagrangian formulation. (3) The throughput of all the methods tested in the paper are on average 25 times slower than that of traditional pressure-correction and velocity-correction methods. Hence, although all these methods are very efficient for solving steady state problems, none of them is unfortunately competitive for solving time-dependent problems.

翻译：本文比较了用于求解非定常不可压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。研究考虑了多种预处理技术（Schur补方法、全耦合系统方法、含/不含增广拉格朗日格式）。重点探究这些方法在计算通量（每核每秒每时间步处理的自由度数量）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，对于大多数方法，GMRES/CG迭代收敛速率与网格尺寸无关，且随雷诺数增加而改善。研究得出三项结论：（1）无论是求解压力Schur补系统还是全耦合系统，在计算通量方面均无显著差异。（2）虽然增广拉格朗日方法展现出优异的并行可扩展性，但最佳计算通量是在不使用增广拉格朗日格式时获得的。（3）本文测试的所有方法的平均计算通量比传统压力修正法和速度修正法慢约25倍。因此，尽管这些方法在求解稳态问题时非常高效，但遗憾的是，在求解非定常问题时均不具备竞争力。