发表机构
The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于笛卡尔网格、易于实现的交错间断伽辽金（SDG）格式，用于Stokes方程，具有压力鲁棒性和二阶超收敛性，并推广至Navier-Stokes方程，实现无条件能量稳定。
AI 中文摘要
本文针对Stokes方程，提出了一种基于笛卡尔网格的交错间断伽辽金（SDG）格式，该格式易于实现，具有内在的压力鲁棒性，并对所有变量均具有超收敛性。与标准SDG中使用的复合网格不同，我们从笛卡尔网格构造交错四边形网格。该格式以速度、压力和速度梯度为未知量，采用分片常数空间，并精心设计了交错连续性。额外的梯度未知量通过静态凝聚在局部消除，并可通过质量集中进一步移除，且不损失精度。我们推导了该格式的显式逐点公式，这便于实现并支持详细的逐点分析。我们严格证明了压力鲁棒性和二阶超收敛性，且该性质在一般非均匀笛卡尔网格上成立。通过引入具有二阶一致性的新型离散对流项，该格式进一步推广至Navier-Stokes方程。结合标量辅助变量（SAV）方法和Crank-Nicolson（CN）格式，我们得到了一个无条件能量稳定且二阶精度的格式。数值实验验证了理论，并展示了其精度和鲁棒性。
英文摘要
This paper develops a staggered discontinuous Galerkin (SDG) scheme based on Cartesian grids for Stokes equations that is simple to implement, intrinsically pressure-robust, and superconvergent for all variables. Instead of the composite meshes used in standard SDG, we construct staggered quadrilateral meshes from Cartesian grids. The scheme takes velocity, pressure, and velocity gradient as unknowns, and employs piecewise-constant spaces with carefully designed staggered continuity. The additional gradient unknowns are locally eliminated via static condensation and can be further removed by mass lumping without loss of accuracy. An explicit pointwise formulation of the scheme is derived, which facilitates implementation and enables a detailed pointwise analysis. We rigorously prove pressure robustness and second-order superconvergence, which holds on general non-uniform Cartesian grids. The scheme is further extended to Navier-Stokes equations by introducing a novel discrete convection term with second-order consistency. Combined with the scalar auxiliary variable (SAV) approach and the Crank-Nicolson (CN) scheme, this yields an unconditionally energy-stable and second-order accurate scheme. Numerical experiments validate the theory and demonstrate accuracy and robustness.
Comments45 pages, 15 figures, 12 tables
