H(curl)-based approximation of the Stokes problem with weakly enforced no-slip boundary conditions

TL;DR

This paper introduces a Nitsche-based finite element method for imposing no-slip boundary conditions in H(curl)-based formulations of the Stokes problem, ensuring stability and optimal convergence.

Contribution

It presents a novel approach to enforce no-slip conditions in H(curl) formulations, overcoming ill-posedness issues and providing rigorous analysis and validation.

Findings

Stable discretization achieved with Nitsche method

Optimal convergence rates demonstrated numerically

Analysis confirms stability and error estimates

Abstract

In this work, we show how to impose no-slip boundary conditions for an H(curl)-based formulation for incompressible Stokes flow, which is used in structure-preserving discretizations of Navier-Stokes and magnetohydrodynamics equations. At first glance, it seems straightforward to apply no-slip boundary conditions: the tangential part is an essential boundary condition on H(curl) and the normal component can be naturally enforced through integration-by-parts of the divergence term. However, we show that this can lead to an ill-posed discretization and propose a Nitsche-based finite element method instead. We analyze the discrete system, establishing stability and deriving a priori error estimates. Numerical experiments validate our analysis and demonstrate optimal convergence rates for the velocity field.