Two-grid finite-element schemes for the transient Navier-Stokes problem

Vivette Girault; Jacques-Louis Lions

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 5, page 945-980
  • ISSN: 0764-583X

Abstract

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We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H . In the second step, the problem is linearized by substituting into the non-linear term, the velocity 𝐮 H computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h . This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of 𝐮 H to the error analysis is measured in the L 2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H 2 . Hence, an error of the order of h can be recovered at the second step, provided h = H 2 .

How to cite

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Girault, Vivette, and Lions, Jacques-Louis. "Two-grid finite-element schemes for the transient Navier-Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.5 (2001): 945-980. <http://eudml.org/doc/194083>.

@article{Girault2001,
abstract = {We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size $H$. In the second step, the problem is linearized by substituting into the non-linear term, the velocity $\{\mathbf \{u\}\}_H$ computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size $h$. This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of $\{\mathbf \{u\}\}_H$ to the error analysis is measured in the $L^2$ norm in space and time, and thus, for the lowest-degree elements, is of the order of $H^2$. Hence, an error of the order of $h$ can be recovered at the second step, provided $h = H^2$.},
author = {Girault, Vivette, Lions, Jacques-Louis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {two grids; a priori estimates; duality; three-dimensional polyhedron; coarse grid; linearized problem; fine grid; error analysis; lowest-degree elements},
language = {eng},
number = {5},
pages = {945-980},
publisher = {EDP-Sciences},
title = {Two-grid finite-element schemes for the transient Navier-Stokes problem},
url = {http://eudml.org/doc/194083},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Girault, Vivette
AU - Lions, Jacques-Louis
TI - Two-grid finite-element schemes for the transient Navier-Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 5
SP - 945
EP - 980
AB - We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size $H$. In the second step, the problem is linearized by substituting into the non-linear term, the velocity ${\mathbf {u}}_H$ computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size $h$. This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of ${\mathbf {u}}_H$ to the error analysis is measured in the $L^2$ norm in space and time, and thus, for the lowest-degree elements, is of the order of $H^2$. Hence, an error of the order of $h$ can be recovered at the second step, provided $h = H^2$.
LA - eng
KW - two grids; a priori estimates; duality; three-dimensional polyhedron; coarse grid; linearized problem; fine grid; error analysis; lowest-degree elements
UR - http://eudml.org/doc/194083
ER -

References

top
  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
  2. [2] A. Ait Ou Amni and M. Marion, Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 62 (1994) 189–213. Zbl0811.76035
  3. [3] D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. Zbl0593.76039
  4. [4] I. Babuška, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179–192. Zbl0258.65108
  5. [5] S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Texts in Applied Mathematics 15, Springer-Verlag, New York (1994). Zbl0804.65101MR1278258
  6. [6] F. Brezzi, On the existence, uniqueness and approximation of saddle-points problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129–151. Zbl0338.90047
  7. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
  8. [8] A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. Zbl0198.50103
  9. [9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). Zbl0383.65058MR520174
  10. [10] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). Zbl0064.33002MR69338
  11. [11] M. Crouzeix, Étude d’une méthode de linéarisation. Résolution numérique des équations de Stokes stationnaires. Application aux équations de Navier-Stokes stationnaires, in Approximation et méthodes itératives de résolution d’inéquations variationnelles et de problèmes non linéaires, in IRIA, Cahier 12, Le Chesnay (1974) 139–244. 
  12. [12] M. Dauge, Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. SIAM J. Math. Anal. 20 (1989) 74–97. Zbl0681.35071
  13. [13] T. Dupont and L.R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34 (1980) 441–463. Zbl0423.65009
  14. [14] C. Foias, O. Manley and R. Temam, Modelization of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO Modél. Anal. Numér. 22 (1988) 93–114. Zbl0663.76054
  15. [15] B. Garcia-Archilla and E. Titi, Postprocessing the Galerkin method: the finite-element case. SIAM J. Numer. Anal. 37 (2000) 470–499. Zbl0952.65078
  16. [16] V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25–57. Zbl0997.76043
  17. [17] V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations, in Lecture Notes in Mathematics 749, Springer-Verlag, Berlin, Heidelberg, New York (1979). Zbl0413.65081MR548867
  18. [18] V. Girault and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms, in Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York (1986). Zbl0585.65077MR851383
  19. [19] R. Glowinski, Finite element methods for the numerical simulation of unsteady incompressible viscous flow modeled by the Navier-Stokes equations. To appear in Handbook of Numerical Analysis, P.G. Ciarlet and J.-L. Lions, Eds., Elsevier, Amsterdam. 
  20. [20] P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24, Pitman, Boston (1985). Zbl0695.35060MR775683
  21. [21] J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980) 639–681. Zbl0494.35077
  22. [22] J. Heywood and R. Rannacher, Finite element approximation of the nonstationnary Navier-Stokes problem. Regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. Zbl0487.76035
  23. [23] O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. In Russian (1961). First English translation, Gordon & Breach, Eds., New York (1963). Zbl0121.42701MR155093
  24. [24] W. Layton, A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 26 (1993) 33–38. Zbl0773.76042
  25. [25] W. Layton and W. Lenferink, Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number. Appl. Math. Comput. 69 (1995) 263–274. Zbl0828.76017
  26. [26] W. Layton and W. Lenferink, A Multilevel mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 17–30. Zbl0844.76053
  27. [27] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12 (1933) 1–82. Zbl0006.16702
  28. [28] J. Leray, Essai sur des mouvements plans d’un liquide visqueux que limitent des parois. J. Math. Pures Appl. 13 (1934) 331–418. Zbl60.0727.01JFM60.0727.01
  29. [29] J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934) 193–248. JFM60.0726.05
  30. [30] J.-L. Lions, Équations différentielles opérationnelles 111. Springer-Verlag, Berlin, Heidelberg, New York (1961). Zbl0098.31101MR153974
  31. [31] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). Zbl0189.40603MR259693
  32. [32] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968). Zbl0165.10801
  33. [33] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Fluids. Oxford University Press, Oxford (1996). Zbl0866.76002MR1422251
  34. [34] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Fluids. Oxford University Press, Oxford (1998). Zbl0908.76004MR1637634
  35. [35] P.-L. Lions, On some challenging problems in nonlinear partial differential equations, in Mathematics: Frontiers and Perspectives; Amer. Math. Soc., Providence, RI (2000) 121–135. Zbl0972.35094
  36. [36] M. Marion and R. Temam, Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26 (1989) 1139–1157. Zbl0683.65083
  37. [37] M. Marion and R. Temam, Nonlinear Galerkin methods: the finite element case. Numer. Math. 57 (1990) 1–22. Zbl0702.65081
  38. [38] M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in Handbook of Numerical Analysis. Vol. VI, P.G. Ciarlet and J.-L. Lions, Eds., Elsevier, Amsterdam (1998) 503–688. Zbl0921.76040
  39. [39] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). MR227584
  40. [40] A. Niemistö, FE-approximation of unconstrained optimal control like problems. Report No. 70. University of Jyväskylä (1995). Zbl0835.65086
  41. [41] O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989). Zbl0712.76001MR1030279
  42. [42] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. Zbl0696.65007
  43. [43] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1979). Zbl0426.35003MR603444
  44. [44] R. Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115–152. Zbl0181.18903
  45. [45] J. Xu, A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994) 231–237. Zbl0795.65077
  46. [46] J. Xu, Two-grid finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33 (1996) 1759–1777. Zbl0860.65119

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