Skip to main content
SpringerLink
Log in

Intrinsic finite element methods for the computation of fluxes for Poisson’s equation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory. We develop a general method for the derivation of intrinsic conforming and non-conforming finite element spaces and appropriate lifting operators for the evaluation of the right-hand side from abstract theoretical principles related to the second Strang Lemma. This intrinsic finite element method is analyzed and convergence with optimal order is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Here, we use the observation that for a polynomial \(q\in \mathbb {P}_{p}\left( \omega \right) \), \(\omega \subset \Omega \) with positive measure, it holds either \(\left. q\right| _{\omega }=0\) or \(\mathrm{supp}\,q=\omega \). In our application we choose \(q=\mathbf {e} _{1}+\mathbf {e}_{2}\) and apply the argument simplex by simplex.

References

  1. Abramowitz, M. , Stegun, I.A.: Handbook of mathematical functions. Applied mathematics series 55. National Bureau of Standards, U.S. Department of Commerce (1972)

  2. Amrouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44(1), 109–140 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Falk, R. S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.), 47(2):281–354 (2010

  4. Baran, Á., Stoyan, G.: Gauss-Legendre elements: a stable, higher order non-conforming finite element family. Computing 79(1), 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi , F., Fortin, M.: Mixed and hybrid finite element methods, vol 15. Springer-Verlag (1991)

  6. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland (1978)

  7. Ciarlet, P.G., Ciarlet Jr, P.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models Methods Appl. Sci. 15(2), 259–271 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ciarlet, P.G., Ciarlet Jr, P.: A new approach for approximating linear elasticity problems. C. R. Math. Acad. Sci. Paris Ser. I 346(5–6), 351–356 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciarlet, P.G., Ciarlet Jr, P.: Direct computation of stresses in planar linearized elasticity. Math. Models Methods Appl. Sci. 19(7), 1043–1064 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Ciarlet, Jr., S. Sauter, and C. Simian. Non-Conforming Intrinsic Finite Elements for Elliptic Boundary Value Problems in Three Dimensions. Technical Report in preparation, 2015

  11. Crouzeix, M., Falk, R.: Nonconforming finite elements for Stokes problems. Math. Comp. 186, 437–456 (1989)

    Article  MathSciNet  Google Scholar 

  12. Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. Revue Française d’Automatique, Informatique et Recherche Opérationnelle 3, 33–75 (1973)

    MathSciNet  Google Scholar 

  13. Fortin, M., Soulie, M.: A nonconforming quadratic finite element on triangles. Int. J. Num. Methods Eng. 19, 505–520 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lee, H., Sheen, D.: Basis for the quadratic nonconforming triangular element of Fortin and Soulie. Int. J. Num. Anal. Model. 2(4), 409–421 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Mardare, S.: On Poincaré and de Rham’s theorems. Revue Roumaine de Mathématiques Pures et Appliquées 53(5–6), 523–541 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Nédélec, J.-C.: Mixed finite elements in R\(^{3}\). Numer. Math. 35(3), 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nédélec, J.-C.: A new family of mixed finite elements in R\(^3\). Numer. Math. 50(1), 57–81 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Heidelberg (2010)

    Book  Google Scholar 

  19. Schwab C.: \(p\)- and \(hp\)-finite element methods. In: Theory and applications in solid and fluid mechanics. The Clarendon Press, Oxford University Press, New York (1998)

  20. Simian C.: Intrinsic discretization of linear elasticity. PhD thesis, University of Zürich (2013)

  21. Stoyan, G., Baran, Á.: Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51(6–7), 967–986 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

    P. G. Ciarlet

  2. Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France

    P. Ciarlet Jr

  3. Institut für Mathematik, Universität Zürich, Winterthurerstr 190, 8057, Zürich, Switzerland

    S. A. Sauter

  4. Department of Computer Science, University of Chicago, 1100 E 58th Street, Chicago, IL, 60637, USA

    C. Simian

Authors
  1. P. G. Ciarlet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Ciarlet Jr
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. A. Sauter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Simian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Ciarlet Jr.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ciarlet, P.G., Ciarlet, P., Sauter, S.A. et al. Intrinsic finite element methods for the computation of fluxes for Poisson’s equation. Numer. Math. 132, 433–462 (2016). https://doi.org/10.1007/s00211-015-0730-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0730-9

Keywords

Mathematics Subject Classification

Search

Navigation