THE NONCONFORMING VIRTUAL ELEMENT METHOD FOR THE STOKES EQUATIONS
THE NONCONFORMING VIRTUAL ELEMENT METHOD FOR THE STOKES EQUATIONS
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DOI：
10.1137/15m1049531
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发表时间：
2016-01-01
影响因子：
2.9
通讯作者：
Manzini, Gianmarco
中科院分区：
文献类型：
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作者：
Cangiani, Andrea;Gyrya, Vitaliy;Manzini, Gianmarco
我们提出了非协调虚拟单元法 (VEM)，用于稳定 Stokes 问题中速度和压力的数值近似。使用不连续分段多项式来近似压力，而使用非相容虚拟元素空间来近似速度的每个分量。在每个网格元素上，局部虚拟空间包含给定次数的多项式空间，以及合适的非多项式函数。虚拟单元函数隐式定义为具有多项式诺伊曼边界条件的局部泊松问题的解。正如 VEM 方法中的典型情况一样，不需要对非多项式函数进行显式评估。这种方法使得可以为任何多项式次数、二维和三维问题以及具有非常一般的多边形和多面体元素的网格构造非相容（虚拟）空间，而不管奇偶性如何。我们证明了不合格 VEM 是 inf-sup 稳定的，并为速度和压力近似建立了最佳先验误差估计。数值例子证实了收敛性分析以及该方法在提供高阶精确近似方面的有效性。
We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.