H(curl)-based approximation of the Stokes problem with slip boundary conditions

Reformulating the incompressible Stokes equations with the velocity sought in H(curl) has recently emerged as a promising approach for the design of helicity-preserving schemes in magnetohydrodynamics and pressure-robust finite element methods on polygonal meshes. A key challenge in this setting, however, is the treatment of Navier slip boundary conditions. In this paper, we overcome this difficulty by recasting the slip condition as a Robin boundary condition and proving well-posedness of the resulting continuous problem. We further identify the geometric and regularity assumptions on the domain and the exact solution under which the classical Stokes solution is recovered. Finally, we study a conforming finite element Galerkin discretization, establishing stability and a priori error estimates. Numerical experiments validate the optimal convergence rates predicted by the theory.

💡 Research Summary

The paper introduces a novel H(curl)-based formulation for the stationary incompressible Stokes problem equipped with Navier slip boundary conditions. Traditional H¹‑based discretizations struggle with slip conditions on general polygonal or polyhedral meshes, especially when pressure‑robustness and helicity preservation are required. By treating the velocity as a differential 1‑form, the authors employ H(curl)‑conforming finite element spaces (e.g., Nédélec elements) to obtain a discretization that naturally respects the underlying geometric structure.

The first technical contribution is the reformulation of the Navier slip condition. Using the result of Mitrea and Monniaux (2009), the classical slip conditions u·n=0 and