Navier-Stokes方程最优控制问题的一种新型投影稳定化方法

本文研究了高雷诺数下二维非定常Navier-Stokes方程最优控制问题的一种新型投影稳定化方法.通过L~2投影稳定化技巧,本文绕开了inf-sup条件对等阶有限元的束缚,克服了雷诺数较大时,对流占优引起的振荡.该方法的优点在于:所有计算只需要在同一套网格上执行,不需要嵌套网格或者将速度和压力的梯度投影到粗网格上.

A new projection method for the optimal control of Navier-Stokes equations

In this paper, a new $L^2$ projection method is proposed for the optimal control of Navier-Stokes equations. The continuous equal-order conforming elements is employed. The method not only overcomes the spurious oscillations due to dominant convection, but also is stable for the equal-order combination of discrete velocity and pressure spaces by adding two local or global $L^2$ projection terms. 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. The stability of the new method is given. For the state, adjoint state and control variables, the a priori error estimates are obtained uniformly with Reynolds number.