Piecewise divergence-free 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. An important property in the analysis is that the local energy projector commutes with the divergence operator. With the help of a divergence-free interpolation operator onto a generalized Raviart-Thomas element space, a pressure-robust nonconforming virtual element method is developed by simply modifying the right hand side of the previous discretization. A reduced virtual element method is also discussed. Numerical results are provided to verify the theoretical convergence.

翻译：用于 Stokes 问题和稳定化的本地能源投影器 之后, 提出了用于 Stokes 问题的无差异和不兼容的虚拟元件方法。 对离散方法进行了详细和严格的误差分析。 分析中的一个重要属性是, 本地能源投影器与差异操作器通通。 在无差异的内插操作器的帮助下, 一个通用的Raviart-Thoomas 元素空间, 一种不兼容的压力- robust 压力- robust 的虚拟元件方法通过简单的修改先前离散法的右侧而得到开发。 也讨论了一个减少的虚拟元件方法。 提供了数字结果以核实理论趋同。