We introduce a novel minimal order hybrid Discontinuous Galerkin (HDG) and a novel mass conserving mixed stress (MCS) method for the approximation of incompressible flows. For this we employ the $H(\operatorname{div})$-conforming linear Brezzi-Douglas-Marini space and the lowest order Raviart-Thomas space for the approximation of the velocity and the vorticity, respectively. Our methods are based on the physically correct diffusive flux $-\nu \varepsilon(u)$ and provide exactly divergence-free discrete velocity solutions, optimal (pressure robust) error estimates and a minimal number of coupling degrees of freedom. For the stability analysis we introduce a new Korn-like inequality for vector-valued element-wise $H^1$ and normal continuous functions. Numerical examples conclude the work where the theoretical findings are validated and the novel methods are compared in terms of condition numbers with respect to discrete stability parameters.

翻译：我们引入了新型最低顺序混合不连续加勒金(HDG)和新型大规模节压混合压力(MCS)近似不可压缩流的方法。 为此,我们采用了$H(Operatorname{div}) 和$($H) 等兼容线线性Brezzi-Douglas-Marini空间,以及拉维亚-Thomas最低顺序空间,分别用于速度和园艺的近似。我们的方法基于物理正确的diffusive通量$-\nu \varepsilon(u) 美元,提供完全不差异的离散速度解决方案、最佳(压力强)误差估计和最小的混合自由度。对于稳定性分析,我们引入了一种新的Korn类不平等,用于矢量值元素值值$H1美元和正常连续函数。数字实例总结了理论结果得到验证和新方法与离散稳定性参数条件数进行比较的工作。