In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.

在本文中，我们介绍了一种新的有限元方法，该方法用于使用H（div）有限元来近似速度来求解一次速度-压力公式中的Stokes方程。像其他通过H（div）服从元素离散速度的有限元方法一样，我们的方法具有精确无散度的速度场和耐压性的优点。但是，大多数H（div）符合Stokes方程的有限元方法需要稳定器来强制切线方向上的速度弱连续性。一些稳定器需要调整惩罚参数，而另一些则不需要。尽管使用了不连续的速度场，但我们的方法没有稳定器。为各种规范中的相应数值近似建立了最佳阶误差估计。进行了广泛的数值研究，以测试该方法的准确性和鲁棒性并确认理论。数值示例涵盖了高达四阶的低阶和高阶近似以及2D和3D情况。