In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization which preserve these invariants at the fully discrete level and we analyse its well-posedness in terms of a CFL condition. Numerical test cases performed with spline finite elements allow us to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

翻译：本文基于de Rham序列中的微分算子及弱形式下具有显式斜对称性的平流算子，提出了两种适用于Navier-Stokes方程的有限元格式。第一种格式通过使用相容性FEEC（有限元外微积分）空间离散该公式得到：它保持了速度的点态无散约束、总动量及能量，同时具有压力鲁棒性。遵循broken-FEEC方法，第二种格式采用全间断空间和局部相容性投影来定义离散微分算子，它通过能量耗散稳定数值间断，从而保持相同的守恒量。针对两种格式，我们采用了中点时间离散化方法，该方法在完全离散水平上保持这些守恒量，并通过CFL条件分析了其适定性。采用样条有限元进行的数值测试案例验证了所得数值方法的高阶精度及其处理一般边界条件的能力。