A non-conforming least-squares finite element method for incompressible fluid flow problems

SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis f...

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Veröffentlicht in:	International journal for numerical methods in fluids Jg. 72; H. 3; S. 375 - 402
Hauptverfasser:	Bochev, Pavel, Lai, James, Olson, Luke
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Bognor Regis Blackwell Publishing Ltd 30.05.2013 Wiley Subscription Services, Inc
Schlagworte:	Boundary conditions Computational fluid dynamics Conservation discontinuous elements Finite element analysis Finite element method Fluid flow Least squares method least-squares finite element methods mass conservation Mathematical analysis Mathematical models piecewise divergence-free velocity pressure Stokes and Navier-Stokes equations Stokes law (fluid mechanics) stream function Studies vorticity
ISSN:	0271-2091, 1097-0363
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Abstract	SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA. We formulate and study numerically a new locally conservative least‐squares FEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and pressure. The method achieves nearly perfect conservation of mass on a series of challenging test problems. Numerical results show excellent agreement with benchmark lid‐driven cavity results. We also propose a simple diagonal pre‐conditioner for the dV–VP formulation, which significantly reduces the condition number of the least‐squares FEM problem.
AbstractList	SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA. We formulate and study numerically a new locally conservative least‐squares FEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and pressure. The method achieves nearly perfect conservation of mass on a series of challenging test problems. Numerical results show excellent agreement with benchmark lid‐driven cavity results. We also propose a simple diagonal pre‐conditioner for the dV–VP formulation, which significantly reduces the condition number of the least‐squares FEM problem. In this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C 0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA. SUMMARY In this paper, we develop least-squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity-vorticity-pressure Stokes system, which uses a piecewise divergence-free basis for the velocity and standard C super(0) elements for the vorticity and the pressure. The new method, which we term dV-VP improves upon our previous discontinuous stream-function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second-order terms from the least-squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one-half of the dimension of a stream-function element of equal accuracy. In two dimensions, the discontinuous stream-function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV-VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA. We formulate and study numerically a new locally conservative least-squares FEM for the velocity-vorticity-pressure Stokes system, which uses a piecewise divergence-free basis for the velocity and standard C super(0) elements for the vorticity and pressure. The method achieves nearly perfect conservation of mass on a series of challenging test problems. Numerical results show excellent agreement with benchmark lid-driven cavity results. We also propose a simple diagonal pre-conditioner for the dV-VP formulation, which significantly reduces the condition number of the least-squares FEM problem. SUMMARY In this paper, we develop least-squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity-vorticity-pressure Stokes system, which uses a piecewise divergence-free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV-VP improves upon our previous discontinuous stream-function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second-order terms from the least-squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one-half of the dimension of a stream-function element of equal accuracy. In two dimensions, the discontinuous stream-function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV-VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA. [PUBLICATION ABSTRACT]
Author	Lai, James Olson, Luke Bochev, Pavel
Author_xml	– sequence: 1 givenname: Pavel surname: Bochev fullname: Bochev, Pavel email: Correspondence to: Pavel Bochev, Sandia National Laboratories, Mail Stop 1320, Albuquerque, NM 87185-1320, USA., pbboche@sandia.gov organization: Numerical Analysis and Applications Department, Sandia National Laboratories, NM, 87185-1320, Mail Stop 1320, Albuquerque, USA – sequence: 2 givenname: James surname: Lai fullname: Lai, James organization: Department of Computer Science, University of Illinois at Urbana-Champaign, IL, 61801, Urbana, USA – sequence: 3 givenname: Luke surname: Olson fullname: Olson, Luke organization: Department of Computer Science, University of Illinois at Urbana-Champaign, IL, 61801, Urbana, USA
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Copyright	Published 2012. This article is a US Government work and is in the public domain in the USA. Copyright © 2013 John Wiley & Sons, Ltd.
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Heroux MA, Bartlett RA, Howle VE, Hoekstra RJ, Hu JJ, Kolda TG, Lehoucq RB, Long KR, Pawlowski RP, Phipps ET, Salinger AG, Thornquist HK, Tuminaro RS, Willenbring JM, Williams A, Stanley KS. An overview of the Trilinos project. ACM Transactions on Mathematical Software 2005; 31(3):397-423. Crouzeix M, Raviart P. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO-Mathematical Modelling and Numerical Analysis 1973; 3:33-76. Proot MM, Gerritsma MI. Mass- and momentum conservation of the least-squares spectral element method for the Stokes problem. Journal of Scientific Computing 2006 June; 27:389-401. Ciarlet P. The Finite Element Method for Elliptic Problems, SIAM Classics in Applied Mathematics. SIAM: Philadelphia, 2002. Kanschat G, Riviere B. A strongly conservative finite element method for the coupling of Stokes and Darcy flow. Journal of Computational Physics 2010; 229(17):5933-5943. Bochev P. 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References_xml	– reference: Bochev P, Choi J. A comparative study of least-squares, SUPG and Galerkin methods for convection problems. International Journal of Fluid Dynamics 2001; 15(2):127-146. – reference: Baker GA, Jureidini WN, Karakashian OA. Piecewise solenoidal vector fields and the Stokes problem. SIAM Journal on Numerical Analysis 1990; 27(6):1466-1485. – reference: Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics 1982; 48(3):387-411. – reference: Crouzeix M, Raviart P. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO-Mathematical Modelling and Numerical Analysis 1973; 3:33-76. – reference: Ciarlet P. The Finite Element Method for Elliptic Problems, SIAM Classics in Applied Mathematics. SIAM: Philadelphia, 2002. – reference: Gunzburger MD. Finite Element Methods for Viscous Incompressible Flows. Academic Press: Boston, 1989. – reference: Pontaza JP, Reddy JN. Spectral/hp least-squares finite element formulation for the Navier-Stokes equations. Journal of Computational Physics 2003; 190(2):523-549. – reference: Karakashian O, Katsaounis T. Numerical simulation of incompressible fluid flow using locally solenoidal elements. Computers & Mathematics with Applications 2006; 51(9-10):1551-1570. – reference: Bolton P, Thatcher RW. On mass conservation in least-squares methods. Journal of Computational Physics 2005; 203(1):287-304. – reference: Bramble JH, Pasciak JE. Least-squares methods for Stokes equations based on a discrete minus one inner product. Journal of Computational and Applied Mathematics 1996; 74(1-2):155-173. – reference: Bazilevs Y, Hughes T. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers & Fluids 2007; 36:12-26. Challenges and Advances in Flow Simulation and Modeling. – reference: Bochev P, Ridzal D, Peterson K. 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Snippet	SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we... In this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we... SUMMARY In this paper, we develop least-squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically,...
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SubjectTerms	Boundary conditions Computational fluid dynamics Conservation discontinuous elements Finite element analysis Finite element method Fluid flow Least squares method least-squares finite element methods mass conservation Mathematical analysis Mathematical models piecewise divergence-free velocity pressure Stokes and Navier-Stokes equations Stokes law (fluid mechanics) stream function Studies vorticity
Title	A non-conforming least-squares finite element method for incompressible fluid flow problems
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