Numerical approximation of generalized Newtonian fluids using Powell–Sabin–Heindl elements: I. theoretical estimates

In this paper we consider the numerical approximation of steady and unsteady generalized Newtonian fluid flows using divergence free finite elements generated by the Powell–Sabin–Heindl elements. We derive a priori and a posteriori finite element error estimates and prove convergence of the method of successive approximations for the steady flow case. A priori error estimates of unsteady flows are also considered. These results provide a theoretical foundation and supporting numerical studies are to be provided in Part II. Copyright © 2003 John Wiley & Sons, Ltd.

[1]  S. Chow.  Finite element error estimates for a blast furnace gas flow problem , 1992 .

[2]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

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

[4]  C. E. Vulliamy.  82. A Note on Proto-Neolithic Flint Implements from the Chiltern Hills. , 1927 .

[5]  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 .

[6]  On the KaČanov method for a quasi-Newtonian flow problem , 1998 .

[7]  Gerhard Heindl,et al.  Interpolation and Approximation by Piecewise Quadratic C1 — Functions of Two Variables , 1979 .

[8]  Jacques Baranger,et al.  Numerical analysis of quasi-Newtonian flow obeying the power low or the Carreau flow , 1990 .

[9]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[10]  Vidar Thomée,et al.  Error estimates for a finite element approximation of a minimal surface , 1975 .

[11]  J. Baranger,et al.  Estimateurs a posteriori d'erreur pour le calcul adaptatif d'écoulements quasi-newtoniens , 1991 .

[12]  J. Baranger,et al.  Analyse numerique des ecoulements quasi-Newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau , 1990 .

[13]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[14]  Dongming Wei,et al.  Existence, uniqueness, and numerical analysis of solutions of a quasilinear parabolic problem , 1992 .