Develop staggered Stokes schemes on general polyhedral meshes

Develop staggered Stokes schemes on general polyhedral meshes by tensorizing the scalar nonparametric finite element spaces for velocity and pairing them with a constant-per-cell pressure approximation space, with the extension intended for time-dependent problems.

Background

The paper develops nonparametric face-based finite element spaces and Poisson schemes for arbitrary polytopal cells, including hexahedral, prismatic, pyramidal, and hybrid three-dimensional meshes. The scalar construction is designed to retain the inclusion of first-degree polynomials and to support optimal convergence on distorted meshes.

The authors identify the extension to staggered Stokes discretizations as the next major research direction. Such an extension would require a tensorized velocity space and a piecewise-constant pressure space, followed by analysis of stability, approximation, consistency, and pressure–velocity coupling, particularly on meshes with curved faces.

References

Although the analysis presented in this article is limited to scalar elliptic operators, the present work lays the foundation for building Stokes elements on general cells. It is indeed possible to build the velocity approximation space by extending the polynomial spaces introduced here tensorially, and using piecewise constant elements for the pressure, as done for the CR and the RT elements; this extension is postponed to future work.

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