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Journal of Scientific Computing

, Volume 69, Issue 3, pp 1192–1250 | Cite as

A Priori and a Posteriori Error Analyses of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

  • Gabriel N. Gatica
  • Filánder A. Sequeira
Article

Abstract

In a recent work we developed a new hybridizable discontinuous Galerkin (HDG) method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The approach there uses the incompressibility condition to eliminate the pressure, sets the gradient of the velocity as an auxiliary unknown, and enriches the original formulation with convenient redundant equations, thus giving rise to an augmented scheme. However, the corresponding analysis, which makes use of a fixed point strategy that depends on a suitably chosen parameter, yields optimal rates of convergence for only two of the six resulting unknowns, whereas the reported numerical results, showing higher orders than predicted, support the conjecture that the a priori error estimates are not sharp. In the present paper, the main features of the aforementioned augmented formulation are maintained, but after introducing slight modifications of the finite element subspaces for the pseudostress and velocity, we are able to significantly improve our previous analyses and results. More precisely, on one hand we realize here that it suffices to choose the stabilization tensor as the identity times the meshsize, and hence neither fixed-point arguments nor related parameters are needed anymore to establish the well-posedness of the discrete scheme, and on the other hand we now prove optimally convergent approximations for all the unknowns. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation of the nonlinear model problem. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the adaptive refinements, are presented.

Keywords

Nonlinear Stokes model Hybridized discontinuous Galerkin method Augmented formulation A posteriori error estimates 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bustinza, R., Gatica, G.N., Cockburn, B.: An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22–23, 147–185 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, H., Li, J., Qiu, W.: Robust a posteriori error estimates for HDG method for convection-diffusion equations. IMA J. Numer. Anal. 36, 437–462 (2016)MathSciNetMATHGoogle Scholar
  7. 7.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  8. 8.
    Clément, P.: Approximation by finite element functions using local regularisation. ESAIM Math. Model. Numer. Anal. 9, 77–84 (1975)MATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.J.: Analysis of HDG methods for Stokes flow. Math. Comp. 80, 723–760 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cockburn, B., Gopalakrishnan, J., Sayas, F.J.: A projection-based error analysis of HDG methods. Math. Comp. 79, 1351–1367 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comp. 78, 1–24 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cockburn, B., Zhang, W.: A posteriori error estimates for HDG methods. J. Sci. Comput. 51, 582–607 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cockburn, B., Zhang, W.: A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 51, 676–693 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Figueroa, L., Gatica, G.N., Márquez, A.: Augmented mixed finite element methods for the stationary Stokes equations. SIAM J. Sci. Comput. 31, 1082–1119 (2008/09)Google Scholar
  15. 15.
    Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, SpringerBriefs in Mathematics. Springer, New York (2014)CrossRefGoogle Scholar
  16. 16.
    Gatica, G.N., Gatica, L.F., Márquez, A.: Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126, 635–677 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gatica, G.N., Gatica, L.F., Sequeira, F.A.: A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity. Comput. Math. Appl. 71, 585–614 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gatica, G.N., Márquez, A., Sánchez, M.A.: Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199, 1064–1079 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefMATHGoogle Scholar
  22. 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)MATHGoogle Scholar
  23. 23.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    Loula, A.F.D., Guerreiro, J.N.C.: Finite element analysis of nonlinear creeping flows. Comput. Methods Appl. Mech. Eng. 99, 87–109 (1990)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199, 582–597 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. (2015). doi: 10.1093/imanum/drv067
  28. 28.
    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). North-Holland, Amsterdam (1991)Google Scholar
  29. 29.
    Sandri, D.: Sur l’approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau. ESAIM Math. Model. Numer. Anal. 27, 131–155 (1993)MathSciNetGoogle Scholar
  30. 30.
    Si, H.: TetGen: A Quality Tetrahedral Mesh Generator and 3D Delaunay Triangulator v.1.5 User’s manual, Tech. Report 13, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2013Google Scholar
  31. 31.
    Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive-Mesh-Refinement Techniques. Wiley, Chichester (1996)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CI2MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Escuela de MatemáticaUniversidad Nacional de Costa RicaHerediaCosta Rica
  3. 3.CI2MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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