This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k − 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence–free.

中文翻译：

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带边界校正的斯托克斯问题的无散有限元方法

本文基于Clough-Tocher分裂上的Scott-Vogelius对构造和分析了Stokes问题的边界修正有限元方法。速度空间由连续的分段多项式组成ķ, 压力空间由次数的分段多项式 (k -1) 没有连续性约束。引入了由关于边界划分的连续分段多项式组成的拉格朗日乘数空间，以强制执行边界条件并减轻压力鲁棒性的不足。我们证明了几个 inf-sup 条件，导致该方法的适定性。此外，我们证明了该方法以最优阶收敛，并且速度近似是无散度的。