In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using serendipity techniques. The main features of the DDR approach are the support of general meshes and arbitrary approximation orders. The method we develop is based on the curl-curl formulation of the momentum equation and, through compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust error estimates for the velocity. It also enables non-standard boundary conditions, such as imposing the value of the pressure on the boundary. In-depth numerical validation on a complete panel of tests including general polyhedral meshes is provided. The paper also contains an appendix where bounds on DDR potential reconstructions and differential operators are proved in the more general framework of Polytopal Exterior Calculus.

翻译：本文设计并分析了一种适用于不可压缩Navier-Stokes方程的离散de Rham（DDR）方法。具体而言，我们聚焦于SDDR变体，该变体通过使用超收敛技术减少了未知量数量。DDR方法的主要特点包括支持一般网格和任意逼近阶数。该方法基于动量方程的旋度-旋度公式，并通过与亥姆霍兹-霍奇分解的兼容性，实现了速度场的压力鲁棒误差估计。此外，该方法还支持非标准边界条件，例如在边界上施加压力值。我们在包含一般多面体网格的完整测试面板上进行了深入的数值验证。论文附录部分还在更广泛的多面体外微分框架中证明了DDR势重建与微分算子的有界性。