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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| Published in: | Communications in numerical methods in engineering Vol. 24; no. 11; pp. 1421 - 1430 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Chichester, UK
John Wiley & Sons, Ltd
01.11.2008
Wiley |
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| 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. |
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| 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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| 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 e_1_2_1_6_2 e_1_2_1_7_2 e_1_2_1_4_2 e_1_2_1_5_2 e_1_2_1_2_2 e_1_2_1_11_2 e_1_2_1_10_2 Brezzi F (e_1_2_1_3_2) 1984 e_1_2_1_8_2 e_1_2_1_9_2 |
| 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 doi: 10.1007/978-3-642-61623-5 – 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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| 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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