Numerical methods for hyperbolic PDEs require stabilization. For linear acoustics, divergence-free vector fields should remain stationary, but classical Finite Difference methods add incompatible diffusion that dramatically restricts the set of discrete stationary states of the numerical method. Compatible diffusion should vanish on stationary states, e.g. should be a gradient of the divergence. Some Finite Element methods allow to naturally embed this grad-div structure, e.g. the SUPG method or OSS. We prove here that the particular discretization associated to them still fails to be constraint preserving. We then introduce a new framework on Cartesian grids based on surface (volume in 3D) integrated operators inspired by Global Flux quadrature and related to mimetic approaches. We are able to construct constraint-compatible stabilization operators (e.g. of SUPG-type) and show that the resulting methods are vorticity-preserving. We show that the Global Flux approach is even super-convergent on stationary states, we characterize the kernels of the discrete operators and we provide projections onto them.

翻译：双曲型偏微分方程的数值方法需要稳定性处理。对于线性声学问题，无散向量场应保持静止，但经典有限差分方法引入了不相容的扩散项，这会严重限制数值方法离散稳态解的集合。相容的扩散项应在稳态解处消失，例如应为散度的梯度。某些有限元方法（如SUPG方法或OSS方法）可自然嵌入这种梯度-散度结构。本文证明与之相关的特定离散格式仍无法保持约束条件。随后，我们基于笛卡尔网格提出新框架，该框架采用受全局通量求积启发并与拟态方法相关的表面积分算子（三维情形为体积分算子）。我们成功构建了约束相容的稳定化算子（如SUPG型），并证明所得方法具有涡量保持特性。研究显示全局通量方法在稳态解上甚至具有超收敛性，我们刻画了离散算子的核空间并给出了向该空间的投影算子。