Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations

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 H1-norm for velocity and the L2-norm for pressure. However, the two-level method involves much less work than the one-level method.

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