The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $P_1-P_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.
翻译:虚拟元素方法(VEM) 是一种Galerkin近似法, 将有限元素法(FEM) 扩展至多面模类。 在本文中, 我们提出了一个符合一致的配方, 将斯托克斯问题数值近似的Scott- Vogelius 限定元素法( Scott- Vogelius) 概括为虚拟元素方法框架内的多边形模类。 特别是, 我们考虑将虚拟元素近似空间直接应用到矢量箱中, 并通过不连续的多面体来近似压力变量。 我们通过调查制造解决方案的趋同性和一组具有代表性的多面形体模类来评估数字近似的效果。 我们用数字显示, 除了三角模类中的最低顺序外, 这个方法与 $P_ 1- P_ 0$ Scott- Vogelius 方案相吻合, 以及方形模类( 这些都是众所周知不稳定的情况) 。