Abstract
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.
Highlights
The Virtual Element Method (VEM) was introduced in Refs. 10 and 12 as a generalization of the Finite Element Method (FEM) allowing for general polytopal meshes
Nowadays the VEM technology has reached a good level of success; among the many papers we here limit ourselves in citing a few sample works.[3,6,11,21,22,34,41,51]
It was soon recognized that the flexibility of VEM allows to build elements that hold peculiar advantages on more standard grids
Summary
The Virtual Element Method (VEM) was introduced in Refs. 10 and 12 as a generalization of the Finite Element Method (FEM) allowing for general polytopal meshes. R −−−i−→ H1(Ω) −−−∇−−→ Σ(Ω) −−c−u−rl→ [H1(Ω)]3 −−d−iv−→ L2(Ω) −−−0−→ 0, with Σ(Ω) denoting functions of L2(Ω) with curl in H1(Ω).[4,43] Discrete Stokes complexes have been extensively studied in the literature of Finite Elements since the presence of an underlying complex implies a series of interesting advantages (such as the divergence-free property), in addition to guaranteeing that the discrete scheme is able to correctly mimic the structure of the problem under study.[7,8,9,27,36,37,39,40,52] This motivation is mainly theoretical in nature, but it serves the important purpose of giving a deeper foundation to our method.
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