A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations

QIN Yan-mei; LI Hui; FENG Min-fu

doi:10.21656/1000-0887.370137

Volume 37 Issue 8

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QIN Yan-mei, LI Hui, FENG Min-fu. A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2016, 37(8): 842-855. doi: 10.21656/1000-0887.370137

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A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations

doi: 10.21656/1000-0887.370137

¹Key Laboratory of Numerical Simulation in the Sichuan Provincial College;School of Maths and Informations Science, Neijiang Normal University, Neijiang, Sichuan 641112, P.R.China;²Sichuan Petroleum and Gas Construction Engineering Co., Ltd., Chengdu 610200, P.R.China;³College of Mathematics, Sichuan University, Chengdu 610064, P.R.China

Funds: The National Natural Science Foundation of China(11271273)

Received Date: 2016-05-05
Rev Recd Date: 2016-06-20
Publish Date: 2016-08-15

Abstract

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.
- Navier-Stokes equation,
- optimal control,
- stabilized method,
- extrapolation formula

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References(27)

References

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