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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,τ 0 ,p H,τ 0 ) on a spatial-time coarse grid J H,τ 0 and a backward Euler solution (u h,τ ,p h,τ ) on a space-time fine grid J h,τ . 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,τ , 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:
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