Navier-Stokes方程的一种压力鲁棒离散de Rham格式
A pressure-robust Discrete de Rham scheme for the Navier-Stokes equations
AI总结：
本文为不可压缩Navier-Stokes方程设计了一种支持一般网格和任意阶近似的SDDR方法，基于curl-curl公式实现压力鲁棒的速度误差估计，并支持非标准边界条件。
摘要：
本文为不可压缩Navier-Stokes方程设计并分析了一种Discrete de Rham (DDR)方法。我们更具体地关注SDDR变体，该变体利用serendipity技术减少了未知数数量。DDR方法的主要特点包括支持一般网格和任意阶近似。我们提出的方法基于动量方程的curl-curl公式，并通过与Helmholtz-Hodge分解的兼容性，为速度提供了压力鲁棒误差估计。该方法还支持非标准边界条件，如在边界上施加压力值。我们在包括一般多面体网格的完整测试集上进行了深入数值验证。论文附录在Polytopal Exterior Calculus的更一般框架下证明了DDR势重构和微分算子的界限。
英文摘要：
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.
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