Design a stable pressure–velocity coupling for curved-face meshes

Design a pressure–velocity coupling bilinear form for staggered Stokes schemes on general polyhedral meshes with curved faces that simultaneously yields locally conservative fluxes across mesh faces, preserves the L2-duality between the discrete velocity-divergence and pressure-gradient operators, and maintains the discrete inf-sup condition despite the use of consistency and stabilization terms.

Background

For non-curved faces, the authors expect the interpolation, velocity-consistency, and pressure–velocity coupling estimates to follow relatively directly from the scalar analysis. Curved faces create an additional difficulty because the pressure–velocity coupling can be expressed either through a discrete velocity divergence or through a discrete pressure gradient.

The desired conservation and duality properties are important for deriving kinetic-energy estimates in time-dependent problems. The authors explicitly note that a naive DG-like stabilization may compromise these properties or violate the inf-sup condition, leaving the appropriate coupling construction unresolved.

References

However, the situation is not as clear in the presence of curved faces. Indeed, the pressure-velocity coupling term can be rewritten, in a finite-volume fashion, as a discrete velocity divergence operator or as a discrete pressure gradient operator. It is desirable to define the discrete velocity divergence operator so that it yields locally conservative fluxes across mesh faces and preserves an $L2$-duality with the discrete pressure gradient operator. These properties are essential, for instance, to derive kinetic energy estimates for time-dependent problems. A naive recovery of consistency through DG-like stabilization terms may break one or both of these properties, or even violate the inf-sup condition. Special care is therefore required to design the velocity-pressure coupling bilinear form.

— Nonparametric face-based finite element methods on complex hybrid three-dimensional meshes  (2610.06208 - Brunel et al., 5 Oct 2026) in Conclusion