This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\boldsymbol{H}({\rm div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\boldsymbol{H}({\bf curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.

翻译：本文包含两项主要贡献。首先，我们遵循离散德拉姆（DDR）和虚拟元（VEM）范式，推导出适用于任意阶次和多面体网格的Stokes方程压力鲁棒方法。与文献中提出的其他方法不同，本文实现压力鲁棒性无需借助子网格上的$\boldsymbol{H}({\rm div})$相容构造，而是将体积力投影到离散$\boldsymbol{H}({\bf curl})$空间。压力误差项的消除源于底层DDR和VEM复形的关键交换性质。我们理论上证明了$h^{k+1}$阶的压力鲁棒误差估计（其中$h$表示网格尺寸，$k\ge 0$表示DDR或VEM复形的多项式次数），并通过一组三维数值实验验证。本文的第二项主要贡献是对DDR与VEM方法之间关系的深入研究。我们特别证明：遵循一种范式开发的复形可以用另一种范式重新表述，且相关联的DDR与VEM复形对在自由度映射下满足交换图性质。