Skip to main content
SpringerLink
Log in

New degrees of freedom for high-order Whitney approximations of Darcy’s flows

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. We propose a high-order discretisation based on Whitney finite elements, namely, Raviart-Thomas finite elements of degree r + 1 for the discharge and discontinuous piecewise polynomial finite elements of degree r for the pressure, with r ≥ 0. We comment on the use of new degrees of freedom that have a clear physical meaning, the so-called weights on the small simplices, for the involved discharge and pressure fields. We describe a new numerical strategy to solve the discrete problem based on a tree-cotree block-decomposition of the unknowns that is natural when considering these new degrees of freedom. Preliminary numerical tests in two dimensions confirm the stability of the adopted method and the effectiveness of the new degrees of freedom.

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

Similar content being viewed by others

References

  1. Alonso Rodríguez, A., Bruni Bruno, L., Rapetti, F.: Minimal sets of unisolvent weights for high order Whitney forms on simplices. Enumath 2019 procs., Lecture Notes in Comput. Sci. and Engng., Vol 139. Springer-Verlag (2020)

  2. Alonso Rodríguez, A., Camaño, J., De Los Santos, E., Rapetti, F.: A graph approach for the construction of high order divergence-free Raviart-Thomas finite elements. Calcolo 55, 42 (2018)

    Article  MathSciNet  Google Scholar 

  3. Alonso Rodríguez, A., Camaño, J., De Los Santos, E., Rapetti, F.: A tree-cotree splitting for the construction of divergence-free finite elements: the high-order case. Submitted hal-02429500 (2019)

  4. Alonso Rodríguez, A., Camaño, J., Ghiloni, R., Valli, A.: Graphs, spanning trees and divergence-free finite elements in domains of general topology. IMA J. Numer. Anal 37, 1986–2003 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Alonso Rodríguez, A., Rapetti, F.: Small trees for high order Whitney elements. Lecture Notes in Comput. Sci. and Engng. 134, 587–597 (2020). Springer-Verlag

    Article  MathSciNet  Google Scholar 

  6. Alotto, P., Perugia, I.: Mixed finite element methods and tree-cotree implicit condensation. Calcolo 36, 233–248 (1999)

    Article  MathSciNet  Google Scholar 

  7. Arnold, D.: Finite element exterior calculus. SIAM isbn: 978-1-611975-53-6 (2018)

  8. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    Article  MathSciNet  Google Scholar 

  9. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. (New Series) of the American Math. Soc. 47, 281–354 (2010)

    Article  MathSciNet  Google Scholar 

  10. Bernardi, C., Maday, Y., Rapetti, F.: Discrétisations variationnelles de problèmes aux limites elliptiques. Mathématiques & Applications 45, 310 (2004). Springer-Verlag

    MATH  Google Scholar 

  11. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Comput. Math. 44, 685 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Bonazzoli, M., Rapetti, F.: High order finite elements in numerical electromagnetism: degrees of freedom and generators in duality. Numer. Algorithms 74, 111–136 (2017)

    Article  MathSciNet  Google Scholar 

  13. Bossavit, A.: Computationalelectromagnetism: Variational Formulations, Complementarity, Edge Elements, p 375. Academic Press, New York (1998)

    Google Scholar 

  14. Chapelle, D., Bathe, K. J.: The inf-sup test. Comput. & Structures 47, 537–545 (1993)

    Article  MathSciNet  Google Scholar 

  15. Christiansen, S. H., Rapetti, F.: On high order finite element spaces of differential forms. Math. Comput. 85, 517–548 (2016)

    Article  MathSciNet  Google Scholar 

  16. Darcy, H.: Les fontaines publiques de la ville de Dijon. In: Dalmont, V. (ed.) Paris, p 647 (1956)

  17. Dubois, F.: Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem. SIAM J. Numer. Anal. 27, 1103–1141 (1990)

    Article  MathSciNet  Google Scholar 

  18. Ern, A., Guermond, J. -L.: Finite elements II: Galerkin Approximation, Elliptic and Mixed PDEs, vol. 73, Texts in Applied Mathematics, Springer (2020)

  19. Fortin, M., Pierre, R.: Stability analysis of discrete generalized stokes problem. Numer. Meth. Partial Differ. Eq. 3, 303–323 (1992)

    Article  MathSciNet  Google Scholar 

  20. Freeze, R. A., Cherry, J. A.: Groundwater, p 604. Prentice Hall, Upper Saddle River (1979)

    Google Scholar 

  21. Hecht, F.: Construction d’une base de fonctions P1 non conforme à divergence nulle dans \(\mathbb {R}^{3}\). RAIRO Anal. Numer. 15, 119–150 (1981)

    Article  MathSciNet  Google Scholar 

  22. Hirani, A. N., Nakshatrala, K. B., Chaudhry, J. H.: Numerical method for Darcy flow derived using discrete exterior calculus. Int. J. Comp. Meth. Engng. Sci. Mech. 16, 151–169 (2015)

    Article  MathSciNet  Google Scholar 

  23. Kuznetsov, N., Nazarov, A.: Sharp constants in the Poincaré, Steklov and related inequalities (a survey). Matematika 1–17 (2014)

  24. Lions, J. L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications I. Springer, New York (1972)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer Series in Comput. Math. 13, 551 (1994)

    MATH  Google Scholar 

  27. Rapetti, F., Bossavit, A.: Whitney forms of higher degree. SIAM Numer. Anal. 47, 2369–2386 (2009)

    Article  MathSciNet  Google Scholar 

  28. Rapetti, F., Dubois, F., Bossavit, A.: Discrete vector potentials for non-simply connected three-dimensional domains. SIAM Numer. Anal. 41(/4), 1505–1527 (2003)

    Article  MathSciNet  Google Scholar 

  29. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of finite element methods. (Proc. Conf. Consiglio Naz. delle Ricerche, Rome, 1975). Lecture Notes in Math., vol. 606, pp 292–315. Springer, Berlin (1977)

    Google Scholar 

  30. Scheichl, R.: Decoupling three-dimensional mixed problems using divergence-free finite elements. SIAM J. Sci. Comput. 23, 1752–1776 (2002)

    Article  MathSciNet  Google Scholar 

  31. Thulasiraman, K., Swamy, M.: Graphs. Theory and Algorithms. A Wiley-Interscience Publication. Wiley, New York (1992)

    Book  Google Scholar 

  32. Tonti, E.: On the mathematical structure of a large class of physical theories. Rend Acc. Lincei 52, 48–56 (1972)

    MathSciNet  Google Scholar 

  33. Whitney, H.: Geometric Integration Theory, p 400. Princeton University Press, Princeton (1957)

    Book  Google Scholar 

Download references

Acknowledgements

The authors wish to dedicate this work to the memory of Christine Bernardi, research director at the CNRS. Numerical methods applied to Darcy’s model was one of her favorite subjects.

Funding

This research was supported by the program MathIT financed by the ANR-15-IDEX-01 of the Universite Côte Azur (UCA) in Nice, France. E. Zappon warmly thanks the Università degli Studi di Trento (Italy) for the possibility of studying as ERASMUS fellow at the Université Côte Azur, where this work began. The first author was partially supported by the project PRIN 201752HKH8.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive, 14, I-38050, Povo, Italy

    Ana Alonso Rodríguez

  2. Laboratoire J.-A. Dieudonné, Université Côte d’Azur, Parc Valrose, F-06108, Nice, France

    Francesca Rapetti

  3. MOX - Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, I-20133, Milano, Italy

    Elena Zappon

Authors
  1. Ana Alonso Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesca Rapetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elena Zappon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesca Rapetti.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alonso Rodríguez, A., Rapetti, F. & Zappon, E. New degrees of freedom for high-order Whitney approximations of Darcy’s flows. Numer Algor 87, 1613–1634 (2021). https://doi.org/10.1007/s11075-020-01022-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-01022-4

Keywords

Mathematics subject classification (2010)

Search

Navigation