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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 $(u_{H,\tau_0},p_{H,\tau_0})$ 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.

76M10Finite element methods (fluid mechanics)
35Q30Stokes and Navier-Stokes equations
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
76D06Statistical solutions of Navier-Stokes and related equations
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