Stabilized Finite Elements for Incompressible, Stationary Navier–Stokes Flows on Manifolds

ABSTRACT A surface finite element method with residual‐based stabilization for stationary Navier–Stokes flows on curved manifolds is introduced. The mixed formulation in stress‐divergence form leads to a system of equations that has a saddle‐point structure. To fulfill the resulting inf‐sup condition, stable element pairs for velocity and pressure are required, for example, Taylor–Hood elements. The Petrov–Galerkin method enables an equal‐order approximation and is developed for flows on surfaces analogously to the classical case in Euclidean space. Additionally, the Lagrange multiplier, which is used to enforce the required tangentiality of the velocity field to the surface, is stabilized based on the residual of the governing differential equations. Furthermore, the SUPG method is adopted for convection‐dominated flows on curved domains. Numerical examples show the application of these residual‐based stabilization methods and confirm their success.