SFEM for the Lagrangian formulation of the surface Stokes problem

Abstract We consider the surface Stokes equation with Lagrange multiplier (L.M.) and approach it numerically. Using a Taylor–Hood surface finite element method, along with an appropriate estimate for the additional L.M., 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 L.M., 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.