Pressure-independent velocity error estimates for (Navier-)Stokes nonconforming virtual element discretization with divergence free

2022 ◽  
Author(s):  
Xin Liu ◽  
Yufeng Nie
Keyword(s):  
Error Estimates ◽  
Navier Stokes ◽  
Element Discretization ◽  
Velocity Error ◽  
Divergence Free ◽  
Pressure Independent ◽  
Virtual Element

Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 353-372 ◽  
Author(s):  
Naveed Ahmed ◽  
Alexander Linke ◽  
Christian Merdon
Keyword(s):  
Mixed Methods ◽  
Stokes Equations ◽  
Mass Conservation ◽  
Approximation Order ◽  
Navier Stokes ◽  
Test Functions ◽  
Navier Stokes Equations ◽  
Velocity Error ◽  
Divergence Free ◽  
Consistency Error

AbstractIn this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy.

scholarly journals A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation

2021 ◽  
Author(s):  
Dibyendu Adak ◽  
David Mora ◽  
Sundararajan Natarajan ◽  
Alberth Silgado
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Virtual Space ◽  
Time Dependent ◽  
Navier Stokes ◽  
Projection Operators ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Element Discretization ◽  
Virtual Element

In this work, a new Virtual Element Method (VEM) of arbitrary order $k \geq 2$ for the time dependent Navier-Stokes equations in stream-function form is proposed and analyzed. Using suitable projection operators, the bilinear and trilinear terms are discretized by only using the proposed degrees of freedom associated with the virtual space. Under certain assumptions on the computational domain, error estimations are derived and shown that the method is optimally convergent in both space and time variables. Finally, to justify the theoretical analysis, four benchmark examples are examined numerically.

scholarly journals Guaranteed Upper Bounds For The Velocity Error Of Pressure-Robust Stokes Discretisations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
P.L. Lederer ◽  
C. Merdon
Keyword(s):  
Mixed Methods ◽  
Error Control ◽  
Stokes Problem ◽  
Upper Bounds ◽  
Type Result ◽  
Numerical Examples ◽  
Velocity Error ◽  
Divergence Free ◽  
Pressure Independent ◽  
Primal And Dual

Abstract This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.

Some Estimates for Virtual Element Methods

2017 ◽  
Vol 17 (4) ◽  
pp. 553-574 ◽  
Author(s):  
Susanne C. Brenner ◽  
Qingguang Guan ◽  
Li-Yeng Sung
Keyword(s):  
Error Estimates ◽  
Three Dimensions ◽  
Poisson Problem ◽  
Polyhedral Meshes ◽  
Virtual Element Methods ◽  
Virtual Element

AbstractWe present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

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