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 two. 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 $L^2$-norm. Numerical experiments are presented, which support the theoretical results.

翻译：本文针对光滑域上的Stokes问题，构建并分析了一类无散有限元方法。离散空间基于任意多项式次数大于二的Scott-Vogelius有限元对。通过结合Piola变换与经典等参框架，并辅以自由度的合理选取，我们证明了该方法在能量范数下具有最优阶收敛性。同时，我们证明了离散速度误差在$L^2$范数下亦达到最优阶收敛。文中给出了数值实验，验证了理论结果。