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

In this paper we study the finite element approximation of systems of $${p(\cdot )}$$p(·)-Stokes type, where $${p(\cdot )}$$p(·) 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.

[1]  On the C1,γ(Ω¯)∩W2,2(Ω) regularity for a class of electro-rheological fluids , 2009 .

[2]  Lars Diening,et al.  Fractional estimates for non-differentiable elliptic systems with general growth , 2008 .

[3]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[4]  Yu. G. Reshetnyak.  Estimates for certain differential operators with finite-dimensional kernel , 1970 .

[5]  M. Ruzicka.  Analysis of Generalized Newtonian Fluids , 2013 .

[6]  W. B. Liu,et al.  Quasi-norm Error Bounds for the Nite Element Approximation of a Non-newtonian Ow , 1994 .

[7]  Kumbakonam R. Rajagopal,et al.  EXISTENCE AND REGULARITY OF SOLUTIONS AND THE STABILITY OF THE REST STATE FOR FLUIDS WITH SHEAR DEPENDENT VISCOSITY , 1995 .

[8]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 2014 .

[9]  D. Breit,et al.  SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs , 2012, 1209.6522.

[10]  John W. Barrett,et al.  Finite element approximation of some degenerate monotone quasilinear elliptic systems , 1996 .

[11]  Lars Diening,et al.  Numerische Mathematik Interpolation operators in Orlicz – Sobolev spaces , 2007 .

[12]  D. Sandri,et al.  Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de carreau , 1993 .

[13]  Franco Brezzi Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics) , 1991 .

[14]  Josef Málek,et al.  On Analysis of Steady Flows of Fluids with Shear-Dependent Viscosity Based on the Lipschitz Truncation Method , 2003, SIAM J. Math. Anal..

[15]  Grigorii Aleksandrovich Seregin,et al.  Regularity for minimizers of some variational problems in plasticity theory , 1993 .

[16]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[17]  John W. Barrett,et al.  A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-newtonian flow , 1998 .

[18]  H. B. Veiga,et al.  Boundary Regularity of Shear Thickening Flows , 2011 .

[19]  L. Diening,et al.  C1,α-regularity for electrorheological fluids in two dimensions , 2007 .

[20]  John W. Barrett,et al.  Finite element approximation of the p-Laplacian , 1993 .

[21]  M. Ruzicka,et al.  An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded , 2001 .

[22]  Giuseppe Mingione,et al.  Regularity Results for Stationary Electro-Rheological Fluids , 2002 .

[23]  J. Málek,et al.  On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications , 2008 .

[24]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[25]  M. Rao,et al.  Theory of Orlicz spaces , 1991 .

[26]  M. Růžička.  Modeling, Mathematical and Numerical Analysis of Electrorheological Fluids , 2004 .

[27]  Lars Diening,et al.  Finite Element Approximation of the p(·)-Laplacian , 2015, SIAM J. Numer. Anal..

[28]  M. Fuchs,et al.  On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids , 2007 .

[29]  J. Málek.  Weak and Measure-valued Solutions to Evolutionary PDEs , 1996 .

[30]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[31]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[32]  川口 光年,et al.  O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Sci. Pub. New York-London, 1963, 184頁, 15×23cm, 3,400円. , 1964 .

[33]  M. Ružička,et al.  Mathematical modeling of electrorheological materials , 2001 .

[34]  Kumbakonam R. Rajagopal,et al.  On the modeling of electrorheological materials , 1996 .

[35]  Andreas Prohl,et al.  Convergence Analysis for Incompressible Generalized Newtonian Fluid Flows with Nonstandard Anisotropic Growth Conditions , 2010, SIAM J. Numer. Anal..

[36]  P. Hästö,et al.  Lebesgue and Sobolev Spaces with Variable Exponents , 2011 .

[37]  M. Fuchs,et al.  A regularity result for stationary electrorheological fluids in two dimensions , 2004 .

[38]  Lars Diening,et al.  On the Finite Element Approximation of p-Stokes Systems , 2012, SIAM J. Numer. Anal..

[39]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .