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Acta Mathematica Vietnamica

, Volume 43, Issue 3, pp 391–413 | Cite as

Discrete Embeddings for Lagrangian and Hamiltonian Systems

  • Jacky Cresson
  • Isabelle Greff
  • Charles Pierre
Article
  • 103 Downloads

Abstract

The topic of this paper is to study the conservation of variational properties for a given problem when discretising it. Precisely, we are interested in Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Consider a partial differential equation (PDE) deriving from a variational principle. A natural question is to know whether this structure is preserved at the discrete level when discretising the PDE. To address this question, a concept of coherence is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. If these procedures finally provide the same discrete problem, we will say that the discretisation is coherent. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations. For Hamiltonian structures, we show the coherence between a discrete Hamiltonian and the discretisation of the mixed formulation of the Poisson problem.

Keywords

PDE discretisation Variational integrators Lagrangian and Hamiltonian systems Discrete embeddings 

Mathematics Subject Classification (2010)

65P10 65M06 65M08 

References

  1. 1.
    Arnol’d, V.I.: Mathematical Methods of Classical Mechanics Graduate Texts in Mathematics, 2nd edn., vol. 60. Springer-Verlag, New York (1989)Google Scholar
  2. 2.
    Brezzi, M., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)CrossRefGoogle Scholar
  4. 4.
    Cresson, J.: Non-differentiable variational principles. J. Math. Anal. Appl. 307 (1), 48–64 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cresson, J., Darses, S.: Stochastic embedding of dynamical systems. J. Math. Phys. 48(7), 54 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cresson, J., Greff, I.: Non-differentiable embedding of Lagrangian systems and partial differential equations. J. Math. Anal. Appl. 384(2), 626–646 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cresson, J., Inizan, P.: Variational formulations of differential equations and asymmetric fractional embedding. J. Math. Anal. Appl. 385(2), 975–997 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crouzeix, M., Raviart, P. -A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R-3), 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Evans, L.C.: Partial Differential Equations Graduate Studies in Mathematics, 2nd edn., vol. 19. American Mathematical Society, Providence (2010)Google Scholar
  10. 10.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis VII, Handb. Numer. Anal., VII, pp. 713–1020. North-Holland (2000)Google Scholar
  11. 11.
    Faou, E.: Geometric Numerical Integration and Schrödinger Equations. Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS). Zürich (2012)Google Scholar
  12. 12.
    Giaquinta, M., Hildebrandt, S.: Calculus of Variations I Grundlehren der Mathematischen Wissenschaften, vol. 310. Springer-Verlag, Berlin (1996)Google Scholar
  13. 13.
    Giaquinta, M., Hildebrandt, S.: Calculus of Variations II Grundlehren der Mathematischen Wissenschaften, vol. 311. Springer-Verlag, Berlin (1996)Google Scholar
  14. 14.
    Girault, V., Raviart, P. -A.: Finite Element Methods for Navier-Stokes Equations Springer Series in Computational Mathematics, vol. 5. Springer-Verlag, Berlin (1986)CrossRefGoogle Scholar
  15. 15.
    Hairer, E.: Important aspects of geometric numerical integration. J. Sci. Comput. 25(1-2), 67–81 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hairer, E.: Challenges in geometric numerical integration. In: Trends in Contemporary Mathematics 8, Springer INdAM Ser., pp. 125–135. Springer, Cham (2014)Google Scholar
  17. 17.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Springer Series in Computational Mathematics, 2nd edn., vol. 31. Springer-Verlag, Berlin (2006)Google Scholar
  18. 18.
    Kane, C., Marsden, J.E., Ortiz, M.: Symplectic-energy-momentum preserving variational integrators. J. Math. Phys. 40(7), 3353–3371 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. ZMM Z. Angew. Math. Mech. 88(9), 677–708 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Raviart, P.-A., Thomas, J.M.: A Mixed Finite Element Method for 2nd Order Elliptic Problems, p. 606. Springer, Berlin (1977)zbMATHGoogle Scholar
  22. 22.
    Thomée, V.: Finite Difference Methods for Linear Parabolic Equations. Handbook of Numerical Analysis I, Handb. Numer. Anal., I, pp. 5–196. North-Holland (1990)Google Scholar
  23. 23.
    Wendlandt, J.M., Marsden, J.E.: Mechanical integrators derived from a discrete variational principle. Phys. D 106(3-4), 223–246 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et de leurs applications, CNRS UMR 5142Université de Pau et des Pays de l’AdourPauFrance

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