Abstract
For the optimal control of Navier-Stokes equations, a new local stabilized nonconforming finite element method was proposed. The time-dependent problem was fully discretized with lowest-equal-order nonconforming finite element NCP<sub>1</sub>-Psub>1</sub>in the velocity and pressure spaces and the reduced Crank-Nicolson scheme in the time domain. The scheme was stable for the equal-order combination of discrete velocity and pressure spaces through the addition of a local L<sup>2</sup> projection term. Specially, based on an extrapolation formula, the method requires only the solution of one linear system per time step. Stability of the method was proved. For the state, adjoint state and control variables, the a priori error estimates were obtained. The error estimation results show that the method has 2nd-order accuracy.
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