SFEM for the Lagrangian formulation of the surface Stokes problem

TL;DR

This paper develops a finite element method for solving the surface Stokes problem with Lagrange multipliers, establishing stability, convergence, and error estimates, supported by numerical simulations and comparison with penalty methods.

Contribution

It introduces a new inf-sup condition for the surface Stokes problem with Lagrange multipliers and proves optimal convergence results, including in standard iso-parametric cases.

Findings

Optimal velocity convergence in energy and tangential L^2 norms

Optimal pressure convergence in L^2 norm

Numerical results confirm theoretical error bounds and compare favorably with penalty methods

Abstract

We consider the surface Stokes equation with Lagrange multiplier and approach it numerically. Using a Taylor-Hood surface finite element method, along with an appropriate estimate for the additional Lagrange multiplier, we derive a new inf-sup condition to help with the stability and convergence results. We establish optimal velocity convergence both in energy and tangential $L^2$ norms, along with optimal $L^2$ norm convergence for the two pressures, in the case of super-parametric finite elements. Furthermore, if the approximation order of the velocities matches that of the extra Lagrange multiplier, we achieve optimal order convergence even in the standard iso-parametric case. In this case, we also establish some new estimates for the normal $L^2$ velocity norm. In addition, we provide numerical simulations that confirm the established error bounds and also perform a comparative analysis against the penalty approach.