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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 $\left({u}_{H,{\tau }_{0}},{p}_{H,{\tau }_{0}}\right)$ on a spatial-time coarse grid ${J}_{H,{\tau }_{0}}$ and a backward Euler solution $\left({u}^{h,\tau },{p}^{h,\tau }\right)$ 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