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 point-wise 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 that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

中文翻译：

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不可压缩纳维-斯托克斯方程的质量、动量和能量守恒 FEEC 和破碎 FEEC 方案


在本文中，我们提出了纳维-斯托克斯方程的两种有限元方案，基于重新表述，其中涉及德拉姆序列的微分算子和弱形式的具有显式斜对称性的平流算子。我们的第一个方案是通过使用符合 FEEC（有限元外微积分）空间对该公式进行离散而获得的：除了具有压力鲁棒性之外，它还保留了速度、总动量和能量的逐点无散度约束。遵循破碎的 FEEC 方法，我们的第二个方案使用完全不连续空间和局部一致投影来定义离散微分算子。它保留相同的不变量直到能量耗散以稳定数值不连续性。对于这两种方案，我们都使用中点时间离散化，将这些不变量保留在完全离散的水平上，并根据 CFL 条件分析其适定性。虽然我们的理论结果适用于保留 de Rham 结构的一般有限元，但我们考虑了张量积样条空间的一种应用。具体来说，我们进行了几个数值测试用例，以验证所得数值方法的高阶精度以及它们处理一般边界条件的能力。