The Divergence-Free Nonconforming Virtual Element for the Stokes Problem

2019 ◽  
Vol 57 (6) ◽  
pp. 2730-2759 ◽  
Author(s):  
Jikun Zhao ◽  
Bei Zhang ◽  
Shipeng Mao ◽  
Shaochun Chen
Keyword(s):  
Stokes Problem ◽  
Divergence Free ◽  
Virtual Element

scholarly journals Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem

2020 ◽  
Author(s):  
Derk Frerichs ◽  
Christian Merdon
Keyword(s):  
Stokes Problem ◽  
Two Dimensions ◽  
Momentum Balance ◽  
Test Functions ◽  
Virtual Test ◽  
Right Hand ◽  
Virtual Element Methods ◽  
Divergence Free ◽  
Virtual Element ◽  
The Right

Abstract Nondivergence-free discretizations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterized by large discretizations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretized in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretization via an $L^2$-best approximation does not preserve the divergence, and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness, another divergence-preserving reconstruction is suggested based on Raviart–Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

scholarly journals Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

2005 ◽  
Vol 75 (254) ◽  
pp. 533-564 ◽  
Author(s):  
Jesús Carrero ◽  
Bernardo Cockburn ◽  
Dominik Schötzau
Keyword(s):  
Stokes Problem ◽  
Divergence Free
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