非定常Oseen方程最优控制问题的一种新型L^2投影稳定化方法

A NEW L^2 PROJECTION METHOD FOR THE OPTIMAL CONTROL OF UNSTEADY OSEEN EQUATIONS

对非定常Oseen方程最优控制问题分析了一种新型L^2投影稳定化方法.空间采用工程上好用的多项式有限元P_l/P_l(l≥1)逼近,时间采用中心差分离散.该稳定化方法对速度和压力分别采用全局或局部L^2投影,不仅绕开了inf-sup条件对等阶元的束缚,而且克服了雷诺数较大,对流占优造成的解的震荡.该方法特点是,所有计算只需要在同一套网格上执行,不需要嵌套的网格或将速度和压力的梯度投影到粗网格上进行计算.给出了详细的误差分析,误差结果与雷诺数一致,且数值解的L^2误差与雷诺数无关.

For the optimal control of unsteady Oseen equations with high Reynolds number, a new L^2 projection method is proposed, where the continuous equal-order conforming elements is employed, and the Crank-Nicolson difference is used for time discretization. Because of adding two local or global L^2 projection terms, the system is not only stable for the equal- order combination of discrete velocity and pressure spaces but also overcomes the spurious oscillations due to dominant convection. Specially, a main advantage of the proposed method is that all the computations are performed at the same element level, without the need of nested meshes or the projection of the gradient of velocity/pressure onto a coarse level. For the state, adjoint state and control variables, the a priori error estimates are obtained uniformly with Reynolds number. Especially the L^2-error estimates of numerical solution are independent of the Reynolds number.