ABSTRACTNumerical 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., there should be a gradient of the divergence. Some Finite Element methods allow the natural embedding of this grad‐div structure, e.g., the SUPG method or OSS. We prove here that the particular discretization associated with them still fails to be stationarity 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 can construct constraint‐compatible stabilization operators (e.g., of SUPG‐type) and show that the resulting methods are stationarity and 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 provide projections onto them.