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On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

Volker John, Alexander Linke, Christian Merdon, Michael Neilan, Leo G. Rebholz
2017 SIAM Review  
The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework.  ...  While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely.  ...  It will be clarified in this review that this lack of robustness is connected with the discretization of the divergence-free constraint in mixed finite element methods.  ... 
doi:10.1137/15m1047696 fatcat:attulpim6fhgrj56y7vqn7cpky

On the divergence constraint in mixed finite element methods for incompressible flows [article]

Volker John, Alexander Linke, Christian Merdon, Michael Neilan, Leo G. Rebholz, University, My, University, My
2019
The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework.  ...  Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of  ...  The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework.  ... 
doi:10.34657/2865 fatcat:u4ly6742rvhxzcp5elwxh3gih4

Page 1461 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
Although the analysis of mixed finite-element methods for three- dimensional incompressible flow problems is well established, this is seemingly not so for the axisymmetric case.  ...  Summary: “The paper presents a finite element method for 3D incompressible fluid flows with capillary free boundaries.  ... 

An explicit divergence-free DG method for incompressible flow

Guosheng Fu
2019 Computer Methods in Applied Mechanics and Engineering  
We present an explicit divergence-free DG method for incompressible flow based on velocity formulation only.  ...  Due to the global nature of the divergence-free constraint, there exist no local bases for our finite element space.  ...  Due to the non-locality of the divergence-free constraint in the finite element space, there exist no local bases.  ... 
doi:10.1016/j.cma.2018.11.012 fatcat:gwf4qs2eljchnjv72dfuny6jzq

Remarks on 4cst-elements for incompressible materials

Noboru Kikuchi
1983 Computer Methods in Applied Mechanics and Engineering  
Three-node triangular elements obtained from the intersecting diagonals of quadrilaterals are thoroughly investigated for the penalty method formulation of incompressible problems such as Stokes flow,  ...  Equivalence of the present formulation to the one using Cnode quadrilateral isoparametric elements with selective reduced integration, is presented.  ...  Acknowledgment The author expresses his sincere thanks to Professor J. Tinsley Oden for his discussions on the subject of the penalty method.  ... 
doi:10.1016/0045-7825(83)90144-5 fatcat:k5xtqk3icrdfnc6riuvqjnycfe

Page 557 of Mathematical Reviews Vol. , Issue 2004a [page]

2004 Mathematical Reviews  
The local approximation within each subdomain is designed using divergence stable /p- mixed finite elements.  ...  We define a two- scale mixed finite element space and resulting method, and describe in detail the solution algorithm.  ... 

An explicit divergence-free DG method for incompressible magnetohydrodynamics [article]

Guosheng Fu
2018 arXiv   pre-print
A globally divergence-free finite element space is used for both the velocity and the magnetic field.  ...  flows so that the CFL constraint is not too restrictive.  ...  The author would like to thank Prof. Chi-Wang Shu for suggesting to work on the problem, and for many helpful discussions concerning the subject.  ... 
arXiv:1808.08119v1 fatcat:cv6zsmcuwfgvnaetela2kxum7y

Page 5743 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
Summary: “A revised mixed finite element method is developed for the quasi-Newtonian flow obeying a power law.  ...  Summary: “We present a numerical scheme for viscoelastic flow based on a second-order central differencing method recently in- troduced in the context of incompressible Newtonian flow; the incompressibility  ... 

Page 6233 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
This paper presents an overview of the finite element method in viscous incompressible flow mostly based on research by the author’s group.  ...  Strikwerda (1-WI-C) 91k:76108 76M10 65N30 76D05 Gresho, Philip M. (1-LLL) The finite element method in viscous incompressible flows.  ... 

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Morten M.T Wang, Tony W.H Sheu
1997 Journal of Computational and Applied Mathematics  
An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows  ...  Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study.  ...  The second source is the absence of pressure in the continuity equation when using mixed formulation methods, While the incompressibility constraint is satisfied a priori in using a mixed finite element  ... 
doi:10.1016/s0377-0427(96)00172-0 fatcat:44zcqynfm5ak7d3qjgrwlljlqy

Page 5055 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
method and mixed finite element methods.  ...  5055 76M Basic methods in fluid mechanics 2001g:76035 76M10 35Q30 65N30 76D07 Gu, Haiming; Liu, Huiming; Liu, Guozhu; Zhao, Sunren New mixed finite element method for Stokes equations.  ... 

Pseudo-divergence-free element free Galerkin method for incompressible fluid flow

Antonio Huerta, Yolanda Vidal, Pierre Villon
2004 Computer Methods in Applied Mechanics and Engineering  
Here, a novel mesh-free formulation is proposed for incompressible flow. It is based on defining a pseudo-divergence-free interpolation space.  ...  Incompressible modelling in finite elements has been a major concern since its early developments and has been extensively studied.  ...  Introduction Accurate and efficient modeling of incompressible flows is an important issue in finite elements. The continuity equation for an incompressible fluid takes the peculiar form.  ... 
doi:10.1016/j.cma.2003.12.010 fatcat:tcqbiol7hjgxdcbnvirzwaod6m

Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem

John A. Evans, Thomas J. R. Hughes
2012 Computational Mechanics  
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining  ...  the data needed, and completing and reviewing the collection of information.  ...  Hughes were partially supported by the Office of Naval Research under contract number N00014-08-0992. John A.  ... 
doi:10.1007/s00466-012-0788-5 fatcat:b5wvqnqukzbt3g5lsxkdxjs6uu

A velocity-stream function method for three-dimensional incompressible fluid flow

Jonas T. Holdeman
2012 Computer Methods in Applied Mechanics and Engineering  
We present a strictly-divergence-free finite element in primitive variables in 3D as the curl of a (generalized) Hermite vector potential element with velocity degrees-of-freedom and show a method for  ...  We describe a velocity-stream function method for computing incompressible fluid flow, extending earlier work in two-to threedimensions.  ...  With this, the finite element presented satisfies the requirements for a truly simple finite element method for incompressible fluid flow.  ... 
doi:10.1016/j.cma.2011.10.013 fatcat:crtoiequ55bvxihglvtizrt4we

Page 5929 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
The second one is Chen-Taylor’s method, which is based on standard bilinear elements for the fluid coupled in a direct conforming way with the solid.  ...  It is based on Raviart-Thomas’ elements for the fluid coupled in a nonconforming way with the elements used for the solid.  ... 
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