This paper has two objectives. On one side, we develop and test numerically divergence-free Virtual Elements in three dimensions, for variable “polynomial” order. These are the natural extension of the two-dimensional divergence-free VEM elements, with some modification that allows for a better computational efficiency. We test the element’s performance both for the Stokes and (diffusion dominated) Navier–Stokes equation. The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex.

中文翻译：

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三维虚拟元素的斯托克斯复合体

本文有两个目标。一方面，我们针对可变“多项式”阶开发和测试三个维度的无数值散度虚拟元素。这些是二维无散度 VEM 元素的自然扩展，经过一些修改以提高计算效率。我们测试了 Stokes 和（以扩散为主的）Navier-Stokes 方程的元素性能。第二个，也许是主要的，动机是表明我们的方案，同样在三个维度上，享有一个潜在的离散斯托克斯复杂结构。我们基于一般多面体分区构建了一对虚拟离散空间，第一个是标量，第二个是向量值，这样当与我们的速度和压力空间相结合时，就会产生离散的斯托克斯复数。