A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations
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摘要: 基于局部Gauss积分和梯形外推公式,速度/压力空间采用最低等阶非协调元NCP1-P1逼近,针对非定常Navier-Stokes方程最优控制问题,建立了一种全离散的非协调有限元局部稳定化格式.该格式绕开了inf-sup条件的束缚,且在每一时间步上,只需要做线性计算,减少了计算量.证明了该格式是无条件稳定的,给出了详细的误差分析.误差结果表明,该线性格式在时间上具有二阶精度.
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关键词:
- Navier-Stokes方程 /
- 最优控制 /
- 稳定化方法 /
- 外推公式
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 NCP1-Psub>1in 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 L2 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.-
Key words:
- Navier-Stokes equation /
- optimal control /
- stabilized method /
- extrapolation formula
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