定常Navier-Stokes方程的三个梯度-散度稳定化Taylor-Hood有限元

Three Grad-Div stabilized Taylor-Hood finite elelments for steady Navier-Stokes eqution

摘要:

本文针对定常Navier-Stokes方程给出了三种梯度-散度稳定化Taylor-Hood元. 为了克服Taylor-Hood混合有限元的离散迭代解不满足质量守恒律的问题，本文在已有的三种迭代格式上增加了梯度-散度稳定项，以便在得到连续离散速度和压力解的同时使离散速度解满足质量守恒律.在强唯一性条件下，本文证明了这三种梯度-散度稳定化Taylor-Hood元迭代格式的离散解在一定迭代次数下逼近Scott-Vogelius 混合有限元离散解.数值实验验证了本文的结果.

Abstract:

We propose three Grad-Div stabilized Taylor-Hood finite elements for the steady Navier-Stokes equation. To keep the law of mass conservation, the Grad-Div stabilized term is added to the known discrete solutions obtained with Taylor-Hood elements, so as to get continuous velocity and pressure, and velocity solutions obeying the law of mass conservation. Under the strong uniqueness conditions, we also show that the Grad-Div stabilized Taylor-Hood finite element iterative solutions converge to the Scott-Vogelius solutions. Finally, numerical examples verify the efficiency of the finifte elements.