Numerische Mathematik

, Volume 132, Issue 4, pp 657–689 | Cite as

Convergence analysis for a finite element approximation of a steady model for electrorheological fluids

  • Luigi C. Berselli
  • Dominic Breit
  • Lars Diening


In this paper we study the finite element approximation of systems of \({p(\cdot )}\)-Stokes type, where \({p(\cdot )}\) is a (non constant) given function of the space variables. We derive—in some cases optimal—error estimates for finite element approximation of the velocity and of the pressure, in a suitable functional setting.

Mathematics Subject Classification

65N15 65N30 35J60 76A05 35Q35 65N12 


  1. 1.
    Acerbi, E., Mingione, G.: Regularity results for stationary electrorheological fluids. Arch. Ration. Mech. Anal 164, 213–259 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bao, W., Barrett, J.W.: A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-Newtonian flow. RAIRO Modél. Math. Anal. Numér. 32, 843–858 (1998)MathSciNetMATHGoogle Scholar
  3. 3.
    Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-Laplacian. Math. Comput. 61(204), 523–537 (1993)MathSciNetMATHGoogle Scholar
  4. 4.
    Barrett, J.W., Liu, W.B.: Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math. 68(4), 437–456 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Belenki, L., Berselli, L.C., Diening, L., Růžička, M.: On the finite element approximation of p-Stokes systems. SIAM J. Numer. Anal. 50(2), 373–397 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bildhauer, M., Fuchs, M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Methods Appl. Sci. 27(13), 1607–1617 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bildhauer, M., Fuchs, M., Zhong, X.: On strong solutions of the differential equations modelling the steady flow of certain incompressible generalized Newtonian fluids. Algebra i Analiz 18, 1–23 (2006). [St. Petersburg Math. J. 18, 183–199 (2007)]Google Scholar
  8. 8.
    Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamic of Polymer Liquids, 2nd edn. Wiley, New York (1987)Google Scholar
  9. 9.
    Beirão da Veiga, H., Kaplický, P., Růžička, M.: Boundary regularity of shear thickening flows. J. Math. Fluid Mech. 13(3), 387–404 (2011)Google Scholar
  10. 10.
    Breit, D.: Smoothness properties of solutions to the nonlinear Stokes problem with non-autonomous potentials. Comment. Math. Univ. Carol. 54, 493–508 (2013)MathSciNetMATHGoogle Scholar
  11. 11.
    Breit, D., Diening, L., Schwarzacher, S.: Solenoidal Lipschitz truncation for parabolic PDEs. Math. Model Methods Appl. Sci. 23, 2671–2700 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Breit, D., Diening, L., Schwarzacher, S.: Finite element methods for the \(p(x)\)-Laplacian. SIAM J. Numer. Anal. 53(1), 551–572 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)Google Scholar
  14. 14.
    Carelli, E., Haehnle, J., Prohl, A.: Convergence analysis for incompressible generalized Newtonian fluid flows with nonstandard anisotropic growth conditions. SIAM. J. Numer. Anal. 48(1), 164–190 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Crispo, F., Grisanti, C.R.: On the \(C^{1,\gamma }(\overline{\Omega })\cap W^{2,2}(\Omega )\) regularity for a class of electro-rheological fluids. J. Math. Anal. Appl. 356(1), 119–132 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Diening, L.: Theoretical and numerical results for electrorheological fluids. Ph.D. thesis, Albert-Ludwigs-Universität, Freiburg (2002)Google Scholar
  17. 17.
    Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math. 20(3), 523–556 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Diening, L., Ettwein, F., Růžička, M.: \(C^{1,\alpha }\)-regularity for electrorheological fluids in two dimensions. NoDEA Nonlinear Differ. Equ. Appl. 14(1–2), 207–217 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Diening, L., Hästö, P., Harjulehto, P., Růžička, M.: Lebesgue and Sobolev spaces with variable exponents. Springer Lecture Notes, vol. 2017. Springer, Berlin (2011)Google Scholar
  20. 20.
    Diening, L., Málek, J., Steinhauer, M.: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim. Calc. Var. 14(2), 211–232 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Diening, L., Růžička, M.: Interpolation operators in Orlicz Sobolev spaces. Numer. Math. 107(1), 107–129 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Frehse, J., Málek, J., Steinhauer, M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34, 1064–1083 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Girault, V., Raviart, P.-A.: Finite element approximation of the Navier–Stokes equations. Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1979)Google Scholar
  24. 24.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York (1969)Google Scholar
  25. 25.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod. Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  26. 26.
    Liu, W.B., Barrett, J.W.: Finite element approximation of some degenerate monotone quasilinear elliptic systems. SIAM J. Numer. Anal. 33(1), 88–106 (1996)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)Google Scholar
  28. 28.
    Málek, J., Rajagopal, K.R., Růžička, M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5, 789–812 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pick, L., Růžička, M.: An example of a space \(l^{p(x)}\) on which the Hardy–Littlewood maximal operator is not bounded. Expo. Math. 19(4), 369–371 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc., New York (1991)Google Scholar
  31. 31.
    Reshetnyak, Y.G.: Estimates for certain differential operators with finite-dimensional kernel. Sib. Math. J. 11, 315–326 (1970)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Rajagopal, K.R., Růžička, M.: On the modeling of electrorheological materials. Mech. Res. Commun. 23, 401–407 (1996)CrossRefMATHGoogle Scholar
  33. 33.
    Rajagopal, K.R., Růžička, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)CrossRefMATHGoogle Scholar
  34. 34.
    Růžička, M.: Electrorheological fluids: modeling and mathematical theory. In: Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)Google Scholar
  35. 35.
    Růžička, M.: Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 49(6), 565–609 (2004)Google Scholar
  36. 36.
    Růžička, M.: Analysis of generalized Newtonian fluids. In: Topics in Mathematical Fluid Mechanics. Lecture Notes in Math., vol. 2073, pp. 199–238. Springer, Heidelberg (2013)Google Scholar
  37. 37.
    Sandri, D.: Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau. RAIRO Modél. Math. Anal. Numér. 27(2), 131–155 (1993)MathSciNetGoogle Scholar
  38. 38.
    Seregin, G.A., Shilkin, T.N.: Regularity of minimizers of some variational problems in plasticity theory. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 243 (1997). [no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii. 28, 270–298, 342–343; translation in J. Math. Sci. New York 99(1), 969–988 (2000)]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Luigi C. Berselli
    • 1
  • Dominic Breit
    • 2
  • Lars Diening
    • 3
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Department of MathematicsHeriot-Watt UniversityRiccarton, EdinburghUK
  3. 3.Institut für MathematikUniversität OsnabrückOsnabrückGermany

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