Navier-Stokes方程最优控制问题的一种非协调有限元局部稳定化方法

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

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.

For the optimal control of N avier-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_1-P_1 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-2 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.