For the discretization of the convective term in the Navier-Stokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D. Sch\"{o}tzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$-conforming divergence-free approximation in CONV. We prove that the modified convective formulation also conserves linear momentum, helicity, 2D enstrophy and total vorticity under some appropriate senses. Its a Picard-type linearization form also conserves them. Under the assumption $\boldsymbol{u}\in L^{2}(0,T;\boldsymbol{W}^{1,\infty}(\Omega)),$ it can be shown that the Gronwall constant does not explicitly depend on the Reynolds number in the error estimates. The long time numerical simulations show that the linearized and modified convective formulation has a similar performance with the EMAC formulation and outperforms the usual skew-symmetric formulation (SKEW).

翻译：对于纳维-斯托克方程式(NSEs)中对流术语的离散化,常用的对流配方(CONV)如果差分限制执行不力,就不会保存能量。在本文中,我们应用了[B. Cockburn, G. Kanschat 和D. Sch\\"{o}tzau, Math. comp. 74(2005), pp. 1067-1095] 的对流化技术,使其符合一定元素方法,恢复了CONV的节能。 关键的想法是用一个 $H(Operatorname{div})$(CONV) 的对离散对流反向速度替换其能量。在CONV中,我们证明修改后的对流相匹配配方也保存了线性动力、外性、外性、2D营养素和整个园艺。 Picard型线性化形式也保存了它们。在假设 $\boldsylsymball{u}(OT;\bold deal demodial demodal demastral demod)上显示了一个不变的公式。