Piecewise divergence-free $H(\textrm{div})$-nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error analysis is presented for the discrete method, including the norm equivalence of the stabilization on the kernel of the local energy projector, the interpolation error estimate, the discrete inf-sup condition, and the optimal error estimate of the discrete method. An important property in the analysis is that the local energy projector commutes with the divergence operator. A reduced virtual element method is also discussed. Numerical results are provided to verify the theoretical convergence.

中文翻译：

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Piecewise Divergence-Free $H(\textrm{div})$ - 任何维度斯托克斯问题的非一致性虚拟元素

Piecewise divergence-free $H(\textrm{div})$-nonconforming virtual elements 是为任何维度的 Stokes 问题设计的。在引入基于斯托克斯问题和镇定性的局部能量投影仪之后，提出了一种无散度的非一致虚元方法来解决斯托克斯问题。对离散方法进行了详细而严格的误差分析，包括局部能量投影器核上的镇定的范数等价性、插值误差估计、离散inf-sup条件和离散方法的最优误差估计. 分析中的一个重要属性是局部能量投影仪与散度算子交换。还讨论了一种简化的虚拟元素方法。提供了数值结果来验证理论收敛性。