Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations.

*(English)* Zbl 1130.76365
Summary: A fully discrete two-level finite element method (the two-level method) is presented for solving the two-dimensional time-dependent Navier-Stokes problem. The method requires a Crank-Nicolson extrapolation solution $({u}_{H,{\tau}_{0}},{p}_{H,{\tau}_{0}})$ on a spatial-time coarse grid ${J}_{H,{\tau}_{0}}$ and a backward Euler solution $({u}^{h,\tau},{p}^{h,\tau})$ on a space-time fine grid ${J}_{h,\tau}$. The error estimates of optimal order of the discrete solution for the two-level method are derived. Compared with the standard Crank-Nicolson extrapolation method (the one-level method) based on a space-time fine grid ${J}_{h,\tau}$, the two-level method is of the error estimates of the same order as the one-level method in the H${}^{1}$-norm for velocity and the L${}^{2}$-norm for pressure. However, the two-level method involves much less work than the one-level method.

##### MSC:

76M10 | Finite element methods (fluid mechanics) |

35Q30 | Stokes and Navier-Stokes equations |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

76D06 | Statistical solutions of Navier-Stokes and related equations |