We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

翻译：我们提出了两种新的最低阶方法——混合方法与混合间断伽辽金法——用于不可压缩流动的逼近。两种方法均采用散度一致的低阶Brezzi-Douglas-Marini空间逼近速度，并采用最低阶Raviart-Thomas空间逼近涡量。我们的方法基于流体物理正确的粘性应力张量（涉及速度的对称梯度而非梯度），能够提供精确无散度的离散速度解，以及具有压力鲁棒性的最优误差估计。我们阐释了如何通过每个面最少的耦合自由度构建这些方法。两种方法的稳定性分析均基于具有连续法向分量的向量有限元的类Korn不等式。数值算例验证了理论结果，并对比了两种新方法的条件数。