Skip to main content
SpringerLink
Log in

Analysis of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving a class of nonlinear Stokes models arising in quasi-Newtonian fluids. Similarly as in previous papers dealing with the application of mixed finite element methods to these nonlinear models, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown. In addition, we enrich the HDG formulation with two suitable augmented equations, which allows us to apply known results from nonlinear functional analysis, namely a nonlinear version of Babuška–Brezzi theory and the classical Banach fixed-point theorem, to prove that the discrete scheme is well-posed and derive the corresponding a priori error estimates. Then we discuss some general aspects concerning the computational implementation of the method, which show a significant reduction of the size of the linear systems involved in the Newton iterations. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation.

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
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Baranger, J., Najib, K., Sandri, D.: Numerical analysis of a three-fields model for a quasi-Newtonian flow. Comput. Methods Appl. Mech. Eng. 109, 281–292 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  3. Bustinza, R., Gatica, G.N.: A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM J. Sci. Comput. 26, 152–177 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bustinza, R., Gatica, G.N.: A mixed local discontinuous Galerkin for a class of nonlinear problems in fluid mechanics. J. Comput. Phys. 207, 427–456 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carrero, J., Cockburn, B., Schötzau, D.: Hybridized, globally divergence-free LDG methods. Part I: the Stokes problem. Math. Comp. 75, 533–563 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Castillo, P.E., Sequeira, F.A.: Computational aspects of the local discontinuous Galerkin method on unstructured grids in three dimensions. Math. Comput. Model. 57, 2279–2288 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Nort-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  8. Clement, P.: Un Problème d’approximation Par éléments Finis, PhD thesis, Ecole Polytechnique Fédérale de Lausane (1973)

  9. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comp. 77, 1887–1916 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80, 723–760 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cockburn, B., Gopalakrishnan, J., Sayas, F.J.: A projection-based error analysis of HDG methods. Math. Comp. 79, 1351–1367 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comp. 78, 1–24 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cockburn, B., Shi, K.: Devising HDG methods for Stokes flow: an overview. Comput. Fluids 98, 221–229 (2014)

    Article  MathSciNet  Google Scholar 

  15. Cockburn, B., Shu, C.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Congreve, S., Houston, P., Süli, E., Whiler, T.P.: Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows. IMA J. Numer. Anal. 33, 1386–1415 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, SpringerBriefs in Mathematics. Springer, Berlin (2014)

    Book  Google Scholar 

  19. Gatica, G.N., González, M., Meddahi, S.: A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. I: a priori error analysis. Comput. Methods Appl. Mech. Eng. 193, 881–892 (2004)

    Article  MATH  Google Scholar 

  20. Gatica, G.N., Heuer, N., Meddahi, S.: On the numerical analysis of nonlinear twofold saddle point problems. IMA J. Numer. Anal. 23, 301–330 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gatica, G.N., Márquez, A., Sánchez, M.A.: A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Comput. Methods Appl. Mech. Eng. 200, 1619–1636 (2011)

    Article  MATH  Google Scholar 

  22. Girault, V., Raviart, P. A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

  23. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Houston, P., Robson, J., Süli, E.: Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems. I. The scalar case. IMA J. Numer. Anal. 25, 726–749 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Howell, J.S.: Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients. J. Comput. Appl. Math. 231, 780–792 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kovasznay, L.I.G.: Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc. 44, 58–62 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ladyzhenskaya, O.: New equations for the description of the viscous incompressible fluids and solvability in the large for the boundary value problems of them. In: Boundary Value Problems of Mathematical Physics V, Providence, RI: AMS (1970)

  28. Loula, A.F.D., Guerreiro, J.N.C.: Finite element analysis of nonlinear creeping flows. Comput. Methods Appl. Mech. Eng. 99, 87–109 (1990)

    Article  MathSciNet  Google Scholar 

  29. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys. 228, 8841–8855 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199, 582–597 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Roberts, J. E., Thomas, J. M.: Mixed and Hybrid Methods. In: Ciarlet, P.G., Lions, J.L. (eds). Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1). Nort-Holland, Amsterdam (1991)

  32. Sandri, D.: Sur l’approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau. Math. Model. Numer. Anal. 27, 131–155 (1993)

    MATH  MathSciNet  Google Scholar 

  33. Schötzau, D., Schwab, C., Toselli, A.: Mixed \(hp\)-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194 (2003)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors are thankful to Paul Castillo and Manuel Solano for valuable remarks concerning the computational implementation of the HDG method.

Author information

Author notes

  1. Filánder A. Sequeira

    Present address: Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Authors and Affiliations

  1. Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica

  2. Escuela de Matemática, Universidad Nacional de Costa Rica, Heredia, Costa Rica

    Filánder A. Sequeira

Authors
  1. Gabriel N. Gatica
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filánder A. Sequeira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel N. Gatica.

Additional information

This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, project Anillo ACT1118 (ANANUM), and the Becas-Chile Programme for foreign students; and by Centro de Investigación en Ingeniería Matemática (\(\hbox {CI}^2\hbox {MA}\)), Universidad de Concepción.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gatica, G.N., Sequeira, F.A. Analysis of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows. J Sci Comput 65, 1270–1308 (2015). https://doi.org/10.1007/s10915-015-0008-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0008-5

Keywords

Search

Navigation