Advertisement

Mimetic Least-Squares Spectral/hp Finite Element Method for the Poisson Equation

  • Artur Palha
  • Marc Gerritsma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5910)

Abstract

Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimise the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme.

Keywords

Little Square Method Dual Variable Spectral Element Spectral Element Method Little Square Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bochev, P., Hyman, J.: Principles of mimetic discretizations of differential operators. IMA 142, 89–119 (2006)MathSciNetGoogle Scholar
  2. 2.
    Bochev, P.: Discourse on variational and geometric aspects of stability of discretizations. 33rd Computational Fluid Dynamics Lecture Series, VKI LS 2003-2005 (2003)Google Scholar
  3. 3.
    Bochev, P., Robinson, A.C.: Matching algorithms with physics: exact sequences of finite element spaces. In: Estep, D., Tavener, S. (eds.) Collected lectures on preservation of stability under discretization. SIAM, PhiladelphiaGoogle Scholar
  4. 4.
    Bossavit, A.: On the geometry of electromagnetism. J. Japan Soc. Appl. Electromagn. & Mech. 6 (1998)Google Scholar
  5. 5.
    Burke, W.L.: Applied differential geometry. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  6. 6.
    Demkowicz, L.: Computing with hp-adaptive finite elements, vol. 1. Chapman and Hall/CRC (2007)Google Scholar
  7. 7.
    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: SIGGRAPH 2005: ACM SIGGRAPH 2005 Courses (2005)Google Scholar
  8. 8.
    Flanders, H.: Differential forms with applications to the physical sciences. Academic Press, Inc., New York (1963)MATHGoogle Scholar
  9. 9.
    Proot, M., Gerritsma, M.: A least-squares spectral element formulation for the Stokes problem. J. Sci. Computing 17, 285–296 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hiptmair, R.: Discrete Hodge operators. Numer. Math. 90, 265–289 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jiang, B.: The least squares finite element method: theory and applications in computational fluid dynamics and electromagnetics. Springer, Heidelberg (1998)MATHGoogle Scholar
  12. 12.
    Kopriva, D.A., Kolias, J.H.: A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows. Journal of Computational Physics 125, 244–261 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mattiussi, C.: An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comp. Physics 133, 289–309 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Proot, M.M.J., Gerritsma, M.I.: Mass- and momentum conservation of the least-squares spectral element method for the Stokes problem. Journal of Scientific Computing 27(1-3), 389–401 (2007)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Tonti, E.: On the formal structure of physical theories. Consiglio Nazionale delle Ricerche, Milano (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Artur Palha
    • 1
  • Marc Gerritsma
    • 1
  1. 1.Delft University of TechnologyDelftThe Netherlands

Personalised recommendations