三维Stokes问题的一种非协调-协调有限元方法

A nonconforming-conforming finite element method for solving 3D Stokes problems

摘要:

本文研究了平行六面体网格下求解三维$Stokes$问题的一种非协调-协调有限元方法。 我们使用非协调旋转$Q_1$元离散速度变量的两个分量，使用协调三线性元离散第三个分量，压力用分片常数离散。 我们证明了该有限元方法满足离散$inf-sup$稳定性条件，且具有最优阶误差估计，即关于速度$\textbf{u}$的一半范和压力$p$的零范一阶收敛。数值试验验证了理论结果。

Abstract:

In this paper, we study nonconforming finite element methods for 3D Stokes problems. We use nonconforming rotated $Q_1$ elements for the approximation of the first two components of the velocity, the conforming trilinear element for the approximation of the third component and piecewise constant for the approximation of pressure. Optimal error estimates are derived, which are both first order for H1-seminorm of velocity $\textbf{u}$ and $L^2$ norm of pressure $p$. Numerical experiments are provided to verify the theoretical results.