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

TL;DR

This paper introduces an H(curl)-based finite element approach for the Stokes problem with slip boundary conditions, providing stability, error estimates, and numerical validation for improved pressure-robust simulations.

Contribution

It reformulates slip boundary conditions as Robin conditions within an H(curl) framework and proves well-posedness, stability, and optimal error estimates for the discretization.

Findings

Numerical experiments confirm optimal convergence rates.

The reformulation effectively handles slip boundary conditions.

The approach enhances pressure-robustness in finite element methods.

Abstract

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.