A simple pressure stabilization method for the Stokes equation

In this paper, we consider a stabilization method for the Stokes problem, using equal‐order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal‐order interpolation in the case of piecewis...

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Vydáno v:Communications in numerical methods in engineering Ročník 24; číslo 11; s. 1421 - 1430
Hlavní autoři: Becker, Roland, Hansbo, Peter
Médium: Journal Article
Jazyk:angličtina
Vydáno: Chichester, UK John Wiley & Sons, Ltd 01.11.2008
Wiley
Témata:
Beräkningsmatematik
Computational Mathematics
Computational techniques
equal-order interpolation
Exact sciences and technology
Fluid dynamics
Fluid Mechanics
Fundamental areas of phenomenology (including applications)
General theory
Mathematical methods in physics
Physics
stabilization
Stokes problem
Strömningsmekanik
Incompressible flow
Stokes problem
Stokes equation
Stabilization
Polynomial approximation
Quadratic approximation
equal-order interpolation
Piecewise linear system
ISSN:1069-8299, 1099-0887, 1099-0887
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Abstract In this paper, we consider a stabilization method for the Stokes problem, using equal‐order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal‐order interpolation in the case of piecewise linear and piecewise quadratic polynomials on triangles. In the case of linear polynomials, we retrieve a well‐known idea of using mass lumping as a stabilization mechanism. Copyright © 2007 John Wiley & Sons, Ltd.
AbstractList In this paper, we consider a stabilization method for the Stokes problem, using equal-order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal-order interpolation in the case of piecewise linear and piecewise quadratic polynomials on triangles. In the case of linear polynomials, we retrieve a well-known idea of using mass lumping as a stabilization mechanism.
In this paper, we consider a stabilization method for the Stokes problem, using equal‐order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal‐order interpolation in the case of piecewise linear and piecewise quadratic polynomials on triangles. In the case of linear polynomials, we retrieve a well‐known idea of using mass lumping as a stabilization mechanism. Copyright © 2007 John Wiley & Sons, Ltd.
Author Hansbo, Peter
Becker, Roland
Author_xml – sequence: 1
  givenname: Roland
  surname: Becker
  fullname: Becker, Roland
  organization: Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l'Adour, BP 1155, 64013 PAU Cedex, France
– sequence: 2
  givenname: Peter
  surname: Hansbo
  fullname: Hansbo, Peter
  email: peter.hansbo@me.chalmers.se
  organization: Division of Computational Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
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Cites_doi 10.1007/s10092-001-8180-4
10.1016/0045-7825(86)90025-3
10.1016/S0045-7825(96)01154-1
10.1002/fld.1650071007
10.1016/j.cma.2005.05.009
10.1002/fld.752
10.1007/978-3-642-61623-5
10.1007/978-1-4612-3172-1
10.1016/j.cam.2007.02.015
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Issue 11
Keywords Incompressible flow
Stokes problem
Stokes equation
Stabilization
Polynomial approximation
Quadratic approximation
equal-order interpolation
Piecewise linear system
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References Girault V, Raviart PA. Finite Elements for the Navier-Stokes Equations. Springer: Berlin, 1986.
Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: New York, 1991.
Codina R, Blasco J. A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Computer Methods in Applied Mechanics and Engineering 1997; 143:373-391.
Löhner R, Morgan K, Peraire J, Vahdati M. Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1987; 7:1093-1109.
Hughes TJR, Franca LP, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation for the Stokes problem accommodating equal order interpolation. Computer Methods in Applied Mechanics and Engineering 1986; 59:89-99.
Dohrmann C, Bochev P. A stabilized finite element method for the Stokes problem based on polynomial pressure projections. International Journal for Numerical Methods in Fluids 2004; 46:183-201.
Li J, He Y. A stabilized finite element method based on two local Gauss integrations for the Stokes equations. Journal of Computational and Applied Mathematics 2007; DOI: 10.1016/j.cam.2007.02.015.
Becker R, Braack M. A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 2001; 38(4):173-199.
Burman E, Hansbo P. Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Computer Methods in Applied Mechanics and Engineering 2006; 195:2393-2410.
1986
2006; 195
2007
2001; 38
1991
1984; 10
1997; 143
1986; 59
2004; 46
1987; 7
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References_xml – reference: Li J, He Y. A stabilized finite element method based on two local Gauss integrations for the Stokes equations. Journal of Computational and Applied Mathematics 2007; DOI: 10.1016/j.cam.2007.02.015.
– reference: Codina R, Blasco J. A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Computer Methods in Applied Mechanics and Engineering 1997; 143:373-391.
– reference: Burman E, Hansbo P. Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Computer Methods in Applied Mechanics and Engineering 2006; 195:2393-2410.
– reference: Löhner R, Morgan K, Peraire J, Vahdati M. Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1987; 7:1093-1109.
– reference: Dohrmann C, Bochev P. A stabilized finite element method for the Stokes problem based on polynomial pressure projections. International Journal for Numerical Methods in Fluids 2004; 46:183-201.
– reference: Girault V, Raviart PA. Finite Elements for the Navier-Stokes Equations. Springer: Berlin, 1986.
– reference: Hughes TJR, Franca LP, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation for the Stokes problem accommodating equal order interpolation. Computer Methods in Applied Mechanics and Engineering 1986; 59:89-99.
– reference: Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: New York, 1991.
– reference: Becker R, Braack M. A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 2001; 38(4):173-199.
– volume: 59
  start-page: 89
  year: 1986
  end-page: 99
  article-title: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation for the Stokes problem accommodating equal order interpolation
  publication-title: Computer Methods in Applied Mechanics and Engineering
– year: 1986
– volume: 38
  start-page: 173
  issue: 4
  year: 2001
  end-page: 199
  article-title: A finite element pressure gradient stabilization for the Stokes equations based on local projections
  publication-title: Calcolo
– volume: 10
  year: 1984
– year: 1991
– volume: 46
  start-page: 183
  year: 2004
  end-page: 201
  article-title: A stabilized finite element method for the Stokes problem based on polynomial pressure projections
  publication-title: International Journal for Numerical Methods in Fluids
– volume: 143
  start-page: 373
  year: 1997
  end-page: 391
  article-title: A finite element formulation for the Stokes problem allowing equal velocity–pressure interpolation
  publication-title: Computer Methods in Applied Mechanics and Engineering
– volume: 195
  start-page: 2393
  year: 2006
  end-page: 2410
  article-title: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method
  publication-title: Computer Methods in Applied Mechanics and Engineering
– volume: 7
  start-page: 1093
  year: 1987
  end-page: 1109
  article-title: Finite element flux‐corrected transport (FEM‐FCT) for the Euler and Navier–Stokes equations
  publication-title: International Journal for Numerical Methods in Fluids
– year: 2007
  article-title: A stabilized finite element method based on two local Gauss integrations for the Stokes equations
  publication-title: Journal of Computational and Applied Mathematics
– ident: e_1_2_1_6_2
  doi: 10.1007/s10092-001-8180-4
– ident: e_1_2_1_4_2
  doi: 10.1016/0045-7825(86)90025-3
– ident: e_1_2_1_5_2
  doi: 10.1016/S0045-7825(96)01154-1
– ident: e_1_2_1_9_2
  doi: 10.1002/fld.1650071007
– ident: e_1_2_1_7_2
  doi: 10.1016/j.cma.2005.05.009
– volume-title: Notes on Numerical Fluid Mechanics, Efficient Solutions of Elliptic Systems
  year: 1984
  ident: e_1_2_1_3_2
– ident: e_1_2_1_8_2
  doi: 10.1002/fld.752
– ident: e_1_2_1_2_2
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– ident: e_1_2_1_11_2
  doi: 10.1007/978-1-4612-3172-1
– ident: e_1_2_1_10_2
  doi: 10.1016/j.cam.2007.02.015
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Snippet In this paper, we consider a stabilization method for the Stokes problem, using equal‐order interpolation of the pressure and velocity, which avoids the use of...
In this paper, we consider a stabilization method for the Stokes problem, using equal-order interpolation of the pressure and velocity, which avoids the use of...
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SubjectTerms Beräkningsmatematik
Computational Mathematics
Computational techniques
equal-order interpolation
Exact sciences and technology
Fluid dynamics
Fluid Mechanics
Fundamental areas of phenomenology (including applications)
General theory
Mathematical methods in physics
Physics
stabilization
Stokes problem
Strömningsmekanik
Title A simple pressure stabilization method for the Stokes equation
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