In this paper, we present a divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes. The velocity and the pressure are discretized by the H(div) virtual element and discontinuous piecewise polynomials, respectively. An additional polynomial space that lives on the element edges is introduced to approximate the tangential trace of the velocity. The velocity at the discrete level is point-wise divergence-free and thus the exact mass conservation is preserved in the discretization. Given suitable data conditions, the well-posedness of the discrete problem is proved and a rigorous error analysis of the method is derived. The error with respect to a mesh dependent norm for the velocity depends on the smoothness of the velocity and the approximation of the load term. A series of numerical experiments are reported to validate the performance o f the method.

在本文中，我们针对多边形网格上的 Navier-Stokes 方程提出了一种无散度的弱虚元方法。速度和压力由H离散化(div) 分别是虚拟元素和不连续分段多项式。引入了位于单元边缘的附加多项式空间来近似速度的切线轨迹。离散水平的速度是无点发散的，因此在离散化中保留了精确的质量守恒。给定合适的数据条件，证明了离散问题的适定性，并导出了该方法的严格误差分析。关于速度的网格相关范数的误差取决于速度的平滑度和载荷项的近似值。报告了一系列数值实验以验证该方法的性能。