Speaker
Description
Surface flow problems have attracted considerable attention in recent years because of their broad applications in science and engineering. In this talk, we investigate the numerical approximation of the surface Stokes equations posed on a two-dimensional manifold embedded in three-dimensional space. The surface velocity is discretized using the nonconforming Crouzeix–Raviart finite element, while the pressure is approximated by piecewise constant functions on a polyhedral approximation of the surface. A fundamental challenge stems from the discretization of the symmetric surface strain-rate tensor, whose kernel contains non-physical velocity modes when combined with the Crouzeix–Raviart element. To address this issue, we introduce a stabilized formulation of the momentum equation and establish the discrete inf-sup stability of the resulting finite element pair with respect to a stabilization-dependent energy norm. We further derive optimal a priori error estimates in both the energy and $L^2$ norms. The analysis is particularly challenging due to the interplay between the symmetric strain-rate tensor, the nonconforming velocity approximation, the pressure variable, and geometric consistency errors arising from the discrete surface. We discuss the key analytical ingredients used to overcome these difficulties and prove the convergence results. Finally, numerical experiments confirm the predicted convergence rates and demonstrate the accuracy and robustness of the proposed method.