A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem

2020 ◽  
Vol 372 ◽  
pp. 113351
Author(s):  
Xin Liu ◽  
Rui Li ◽  
Yufeng Nie
Keyword(s):  
Stokes Problem ◽  
Virtual Element Method ◽  
Divergence Free ◽  
Virtual Element ◽  
Element Method

scholarly journals An H1-conforming virtual element for Darcy and Brinkman equations

2017 ◽  
Vol 28 (01) ◽  
pp. 159-194 ◽  
Author(s):  
Giuseppe Vacca
Keyword(s):  
Stokes Equations ◽  
Brinkman Equation ◽  
Optimal Order ◽  
Virtual Element Method ◽  
Element Space ◽  
Optimal Order Of Convergence ◽  
Brinkman Equations ◽  
Divergence Free ◽  
Virtual Element ◽  
Element Method

The focus of this paper is on developing a virtual element method (VEM) for Darcy and Brinkman equations. In [L. Beirão da Veiga, C. Lovadina and G. Vacca, ESAIM Math. Model. Numer. Anal. 51 (2017)], we presented a family of virtual elements for Stokes equations and we defined a new virtual element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different virtual element space having two fundamental properties: the [Formula: see text]-projection onto [Formula: see text] is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and [Formula: see text]-conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.

The nonconforming virtual element method for the Darcy–Stokes problem

2020 ◽  
Vol 370 ◽  
pp. 113251
Author(s):  
Jikun Zhao ◽  
Bei Zhang ◽  
Shipeng Mao ◽  
Shaochun Chen
Keyword(s):  
Stokes Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Gianmarco Manzini ◽  
Annamaria Mazzia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Approximation ◽  
Convergence Rates ◽  
Stokes Problem ◽  
Element Approximation ◽  
Virtual Element Method ◽  
Polygonal Meshes ◽  
Virtual Element ◽  
Element Method

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $\end{document}</tex-math></inline-formula> Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.</p>

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