In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The backward Euler scheme is considered for the temporal discretization, and two-grid method is used for the space discretization. The key idea is that for a solution to the unsteady Navier-Stokes problem, we could use a relatively coarse mesh to approximate low-frequency components and use some local fine mesh to compute high-frequency components. Some local a priori estimate is obtained. With that, theoretical results are derived. Finally, some numerical results are reported to support the theoretical findings.

针对非定常的Navier-Stokes方程，提出并研究了基于二重网格离散化的局部和并行有限元方法。对于时间离散化，考虑了向后欧拉方案，而对于空间离散化，则采用了两网格方法。关键思想是，对于不稳定的Navier-Stokes问题的解决方案，我们可以使用相对粗糙的网格来近似低频分量，并使用一些局部精细网格来计算高频分量。获得一些局部先验估计。这样，得出了理论结果。最后，报告了一些数值结果以支持理论发现。