H(div) conforming methods for the rotation form of the incompressible fluid equations

Abstract

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 Guzman 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 Guzman 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.