A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem

2020 ◽  
Vol 85 (1) ◽  
Author(s):  
Jun Guo ◽  
Minfu Feng
Keyword(s):  
Stokes Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

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>

scholarly journals Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

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