A General Degree Divergence-Free Finite Element Method for the Two-Dimensional Stokes Problem on Smooth Domains

In this paper, we construct and analyze divergence-free finite element methods for the Stokes problem on smooth domains. The discrete spaces are based on the Scott-Vogelius finite element pair of arbitrary polynomial degree greater than 2. By combining the Piola transform with the classical isoparametric framework, and with a judicious choice of degrees of freedom, we prove that the method converges with optimal order in the energy norm. We also show that the discrete velocity error converges with optimal order in the \(L^2\) $L^2$ -norm. Numerical experiments are presented, which support the theoretical results.