New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzmán et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the incompressible Navier-Stokes equation with the full version of the stress tensor \(\nu \left( \nabla \varvec{u}+ \nabla \varvec{u}^T - \frac{2}{3} \left( \nabla \cdot \varvec{u}\right) \mathbb {I} \right)\), instead of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzmán et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order \(p\in \{2,3\}\), (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization.

中文翻译：

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不可压缩流体方程旋转形式的H（div）拟合方法

针对不可压缩流设计了新的H（div）相容有限元方法，该方法涉及运动方程和Bernoulli函数的旋转形式。通过选择特定的数值通量，我们恢复了与Guzmán等人相同的速度场。（IMA J Numer Anal 37（4）：1733–1771，2016），用于对流形式的不可压缩的欧拉方程。误差估计是针对半离散方法提出的。我们用应力张量的完整版本\（\ nu \ left（\ nabla \ varvec {u} + \ nabla \ varvec {u} ^ T-\ frac {2} {3}进一步研究不可压缩的Navier-Stokes方程\ left（\ nabla \ cdot \ varvec {u} \ right）\ mathbb {I} \ right）\），而不是在连续级别上部分执行无散度约束（如在有限元方法中通常执行的那样），我们让数值方案完全控制该约束的执行。最后，我们通过一些数值模拟测试了所提出方法的行为。我们的结果表明（1）在古兹曼（Guzmán）等人中，我们恢复了相同的速度场。（2016），（2）当H（div）符合BDM-DG元素时，与Schroeder等人相比，我们获得的速度误差较小。（SeMA J 75（4）：629–653，2018）多项式阶数\（p \ in \ {2,3 \} \），（3）当H1符合Taylor-Hood元素时，使用全应力张量有助于减少速度和伯努利函数的误差，（4）H（div）符合方法在长期结构保存方面做得更好与经典的混合方法相比，甚至具有grad-div稳定。