On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes
SUMMARY The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary incompressible flows. The observed behavior is due to the fact that discrete solenoidal fields lose this property under changes of the spatial...
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| Published in | International journal for numerical methods in fluids Vol. 69; no. 6; pp. 1045 - 1064 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
30.06.2012
Wiley |
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| Abstract | SUMMARY
The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary incompressible flows. The observed behavior is due to the fact that discrete solenoidal fields lose this property under changes of the spatial discretization. This phenomenon is analyzed for DG finite element discretizations in time, and possible ways are considered to circumvent this problem. Copyright © 2011 John Wiley & Sons, Ltd. |
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| AbstractList | The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary incompressible flows. The observed behavior is due to the fact that discrete solenoidal fields lose this property under changes of the spatial discretization. This phenomenon is analyzed for DG finite element discretizations in time, and possible ways are considered to circumvent this problem. Copyright © 2011 John Wiley & Sons, Ltd. SUMMARY The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary incompressible flows. The observed behavior is due to the fact that discrete solenoidal fields lose this property under changes of the spatial discretization. This phenomenon is analyzed for DG finite element discretizations in time, and possible ways are considered to circumvent this problem. Copyright © 2011 John Wiley & Sons, Ltd. |
| Author | Wollner, Winnifried Besier, Michael |
| Author_xml | – sequence: 1 givenname: Michael surname: Besier fullname: Besier, Michael email: michael.besier@iwr.uni-heidelberg.de, Michael Besier, Department of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 293/294, 69120 Heidelberg, Germany., michael.besier@iwr.uni-heidelberg.de organization: Department of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 293/294, 69120, Heidelberg, Germany – sequence: 2 givenname: Winnifried surname: Wollner fullname: Wollner, Winnifried organization: Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146, Hamburg, Germany |
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| Keywords | Computational fluid dynamics space-time finite elements Digital simulation Unsteady flow Finite element method incompressible Navier―Stokes equations DG method in time dynamically changing meshes Modelling backward Euler scheme Incompressible fluid pressure approximation Mesh generation Navier-Stokes equations |
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| References | Temam R. Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing: Providence, Rhode Island, 2001. Thomée V. Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, Vol. 25. Springer-Verlag: Berlin Heidelberg, 1997. Kellogg RB, Osborn JE. A regularity result for the Stokes problem in a convex polygon. Journal of Functional Analysis 1976; 21(4): 397-431. Klouček P, Rys FS. Stability of the fractional step θ-scheme for the nonstationary Navier-Stokes equations. SIAM Journal on Numerical Analysis 1994; 31(5): 1312-1335. Dauge M. Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations. SIAM Journal on Mathematical Analysis 1989; 20(1): 74-97. Turek S. Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach (Lecture Notes in Computational Science and Engineering), Vol. 6. Springer-Verlag: Berlin Heidelberg, 1999. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15. Springer-Verlag: New York Berlin Heidelberg, 1991. Becker R, Braack M. A modification of the least-squares stabilization for the Stokes equations. Calcolo 2001; 38(4): 173-199. Heywood JG, Rannacher R, Turek S. Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1996; 22(5): 325-352. Bristeau MO, Glowinski R, Periaux J. Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incomp ressible viscous flows. Computer Physics Reports 1987; 6(1-6): 73-187. Turek S.A comparative study of time-stepping techniques for the incompressible Navier-Stokes equations: From fully implicit non-linear schemes to semi-implicit projection methods. International Journal for Numerical Methods in Fluids 1996; 22(10): 987-1011. Bänsch E. An adaptive finite-element strategy for the three-dimensional time-dependent Navier-Stokes equations. Journal of Computational and Applied Mathematics 1991; 36(1): 3-28. Girault V, Raviart P-A. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag: Berlin Heidelberg New York Tokyo, 1986. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner Series Advances in Numerical Mathematics. Wiley-Teubner: New York Stuttgart, 1996. Ciarlet PG. The Finite Element Method for Elliptic Problems, First edn, North-Holland Publishing Company: Amsterdam New York Oxford, 1987. Heywood JG, Rannacher R. Finite element approximation of the nonstationary Navier-Stokes problem. Part I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM Journal on Numerical Analysis 1982; 19(2): 275-311. John V. Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder. International Journal for Numerical Methods in Fluids 2004; 44(7): 777-788. 1976; 21 2004; 44 1991; 36 1989; 20 1991; 15 1982; 19 2010 1997; 25 1987; 6 2009 1998 1996; 52 1996 1974 2004 1995; 4 1999; 6 2001 1986; 5 2003; 9 1987 2003; 25 2001; 38 1996; 22 1994; 31 e_1_2_8_28_1 e_1_2_8_29_1 e_1_2_8_24_1 e_1_2_8_25_1 e_1_2_8_26_1 e_1_2_8_27_1 Ciarlet PG (e_1_2_8_11_1) 1987 e_1_2_8_3_1 Hoffman J (e_1_2_8_6_1) 2003 e_1_2_8_2_1 e_1_2_8_4_1 e_1_2_8_7_1 e_1_2_8_9_1 e_1_2_8_8_1 e_1_2_8_20_1 Hood P (e_1_2_8_13_1) 1974 e_1_2_8_21_1 e_1_2_8_22_1 e_1_2_8_23_1 e_1_2_8_17_1 e_1_2_8_18_1 e_1_2_8_19_1 e_1_2_8_14_1 e_1_2_8_15_1 e_1_2_8_16_1 Verfürth R (e_1_2_8_5_1) 1996 e_1_2_8_10_1 e_1_2_8_12_1 |
| References_xml | – reference: Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15. Springer-Verlag: New York Berlin Heidelberg, 1991. – reference: Bristeau MO, Glowinski R, Periaux J. Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incomp ressible viscous flows. Computer Physics Reports 1987; 6(1-6): 73-187. – reference: Dauge M. Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations. SIAM Journal on Mathematical Analysis 1989; 20(1): 74-97. – reference: Heywood JG, Rannacher R. Finite element approximation of the nonstationary Navier-Stokes problem. Part I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM Journal on Numerical Analysis 1982; 19(2): 275-311. – reference: John V. Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder. International Journal for Numerical Methods in Fluids 2004; 44(7): 777-788. – reference: Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner Series Advances in Numerical Mathematics. Wiley-Teubner: New York Stuttgart, 1996. – reference: Klouček P, Rys FS. Stability of the fractional step θ-scheme for the nonstationary Navier-Stokes equations. SIAM Journal on Numerical Analysis 1994; 31(5): 1312-1335. – reference: Temam R. Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing: Providence, Rhode Island, 2001. – reference: Thomée V. Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, Vol. 25. Springer-Verlag: Berlin Heidelberg, 1997. – reference: Bänsch E. An adaptive finite-element strategy for the three-dimensional time-dependent Navier-Stokes equations. Journal of Computational and Applied Mathematics 1991; 36(1): 3-28. – reference: Girault V, Raviart P-A. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag: Berlin Heidelberg New York Tokyo, 1986. – reference: Ciarlet PG. The Finite Element Method for Elliptic Problems, First edn, North-Holland Publishing Company: Amsterdam New York Oxford, 1987. – reference: Kellogg RB, Osborn JE. A regularity result for the Stokes problem in a convex polygon. Journal of Functional Analysis 1976; 21(4): 397-431. – reference: Becker R, Braack M. A modification of the least-squares stabilization for the Stokes equations. Calcolo 2001; 38(4): 173-199. – reference: Heywood JG, Rannacher R, Turek S. Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1996; 22(5): 325-352. – reference: Turek S. Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach (Lecture Notes in Computational Science and Engineering), Vol. 6. Springer-Verlag: Berlin Heidelberg, 1999. – reference: Turek S.A comparative study of time-stepping techniques for the incompressible Navier-Stokes equations: From fully implicit non-linear schemes to semi-implicit projection methods. International Journal for Numerical Methods in Fluids 1996; 22(10): 987-1011. – start-page: 121 year: 1974 end-page: 132 – year: 2009 – volume: 44 start-page: 777 issue: 7 year: 2004 end-page: 788 article-title: Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder publication-title: International Journal for Numerical Methods in Fluids – volume: 38 start-page: 173 issue: 4 year: 2001 end-page: 199 article-title: A modification of the least‐squares stabilization for the Stokes equations publication-title: Calcolo – volume: 22 start-page: 987 issue: 10 year: 1996 end-page: 1011 article-title: A comparative study of time‐stepping techniques for the incompressible Navier–Stokes equations: From fully implicit non‐linear schemes to semi‐implicit projection methods publication-title: International Journal for Numerical Methods in Fluids – volume: 4 start-page: 105 year: 1995 end-page: 158 – year: 2001 – year: 1987 – volume: 22 start-page: 325 issue: 5 year: 1996 end-page: 352 article-title: Artificial boundaries and flux and pressure conditions for the incompressible Navier‐Stokes equations publication-title: International Journal for Numerical Methods in Fluids – year: 1996 – volume: 25 start-page: 97 year: 2003 end-page: 158 – volume: 19 start-page: 275 issue: 2 year: 1982 end-page: 311 article-title: Finite element approximation of the nonstationary Navier‐Stokes problem. Part I: Regularity of solutions and second‐order error estimates for spatial discretization publication-title: SIAM Journal on Numerical Analysis – volume: 15 year: 1991 – volume: 6 start-page: 73 issue: 1–6 year: 1987 end-page: 187 article-title: Numerical methods for the Navier‐Stokes equations. Applications to the simulation of compressible and incomp ressible viscous flows publication-title: Computer Physics Reports – year: 2010 – year: 1998 – volume: 25 year: 1997 – volume: 52 start-page: 547 year: 1996 end-page: 566 – volume: 20 start-page: 74 issue: 1 year: 1989 end-page: 97 article-title: Stationary Stokes and Navier‐Stokes systems on two‐ or three‐dimensional domains with corners. Part I: Linearized equations publication-title: SIAM Journal on Mathematical Analysis – volume: 36 start-page: 3 issue: 1 year: 1991 end-page: 28 article-title: An adaptive finite‐element strategy for the three‐dimensional time‐dependent Navier‐Stokes equations publication-title: Journal of Computational and Applied Mathematics – volume: 5 year: 1986 – volume: 9 start-page: 3 year: 2003 end-page: 1776 – volume: 21 start-page: 397 issue: 4 year: 1976 end-page: 431 article-title: A regularity result for the Stokes problem in a convex polygon publication-title: Journal of Functional Analysis – start-page: 123 year: 2004 end-page: 130 – volume: 6 year: 1999 – volume: 31 start-page: 1312 issue: 5 year: 1994 end-page: 1335 article-title: Stability of the fractional step ‐scheme for the nonstationary Navier‐Stokes equations publication-title: SIAM Journal on Numerical Analysis – start-page: 121 volume-title: Finite Element Methods in Flow Problems year: 1974 ident: e_1_2_8_13_1 – volume-title: The Finite Element Method for Elliptic Problems year: 1987 ident: e_1_2_8_11_1 – ident: e_1_2_8_22_1 doi: 10.1016/0167-7977(87)90011-6 – ident: e_1_2_8_20_1 – ident: e_1_2_8_27_1 doi: 10.1016/0022-1236(76)90035-5 – ident: e_1_2_8_17_1 – ident: e_1_2_8_12_1 doi: 10.1007/978-3-642-61623-5 – ident: e_1_2_8_15_1 doi: 10.1007/s10092-001-8180-4 – ident: e_1_2_8_21_1 doi: 10.1137/0719018 – ident: e_1_2_8_24_1 doi: 10.1137/0731068 – ident: e_1_2_8_16_1 doi: 10.1007/978-3-642-18775-9_9 – ident: e_1_2_8_10_1 doi: 10.1007/978-3-662-03359-3 – ident: e_1_2_8_7_1 – start-page: 97 volume-title: Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Lecture Notes in Computational Science and Engineering year: 2003 ident: e_1_2_8_6_1 – ident: e_1_2_8_25_1 doi: 10.1002/(SICI)1097-0363(19960530)22:10<987::AID-FLD394>3.0.CO;2-7 – ident: e_1_2_8_2_1 doi: 10.1007/978-3-322-89849-4_39 – ident: e_1_2_8_9_1 doi: 10.1090/chel/343 – ident: e_1_2_8_29_1 doi: 10.1002/fld.679 – ident: e_1_2_8_4_1 doi: 10.1016/0377-0427(91)90224-8 – ident: e_1_2_8_26_1 doi: 10.1007/978-3-642-58393-3 – ident: e_1_2_8_8_1 – ident: e_1_2_8_18_1 – ident: e_1_2_8_23_1 doi: 10.1016/S1570-8659(03)09003-3 – ident: e_1_2_8_3_1 doi: 10.1017/S0962492900002531 – volume-title: A Review of A Posteriori Error Estimation and Adaptive Mesh‐Refinement Techniques year: 1996 ident: e_1_2_8_5_1 – ident: e_1_2_8_19_1 doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y – ident: e_1_2_8_28_1 doi: 10.1137/0520006 – ident: e_1_2_8_14_1 doi: 10.1007/978-1-4612-3172-1 |
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The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary... The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary... |
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| SubjectTerms | backward Euler scheme Computational methods in fluid dynamics DG method in time dynamically changing meshes Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) incompressible Navier-Stokes equations Physics pressure approximation space-time finite elements |
| Title | On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes |
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