A posteriori estimators for mixed finite element approximations of a fluid obeying the power law. (English) Zbl 0953.76057
Summary: We study a posteriori error estimation for the approximation by finite element method of a fluid which obeys the power law: \(-2\mu\nabla\cdot (|d(u)|^{r- 2}d(u))+ \nabla p=f\), \(\nabla u= 0\), \(1< r< 2\). Abstract and calculable a posteriori estimators are given in two-fields and in a three-fields version of this problem. Furthermore, we give some examples of finite element spaces satisfying the inf-sup condition, which relates the discrete space of tensors and the discrete space of velocities, needed for this formulation.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76A05 | Non-Newtonian fluids |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
velocity-pressure formulation; finite element spaces; Carreau law; tensor-velocity-pressure formulation; power law fluid; a posteriori error estimation; finite element method; inf-sup condition; discrete space of tensors; discrete space of velocitiesReferences:
| [1] | Amrouche, C.; Girault, V., Propriétés fonctionnelles d’opérateurs. Application au problème de Stokes en dimension quelconque, (Publication du Laboratoire d’Analyse Numérique, no 90025 (1990), Université Pierre et Marie Curie: Université Pierre et Marie Curie Paris, France) |
| [2] | Baranger, J.; Amri, H. El, Estimateurs a posteriori d’erreur pour le calcul adaptatif d’écoulements quasi-Newtoniens, \(M^2\) AN, 25, 1, 31-48 (1991) · Zbl 0712.76068 |
| [3] | Baranger, J.; Georget, P.; Najib, K., Error estimates for a mixed finite element method for a non Newtonian flow, J. Non-Newt. Fluid Mech., 23, 415-421 (1987) · Zbl 0619.76003 |
| [4] | Baranger, J.; Najib, K., Analyse numérique des écoulements quasi-Newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau, Numer. Math., 58, 35-49 (1990) · Zbl 0702.76007 |
| [5] | Baranger, J.; Najib, K.; Sandri, D., Numerical analysis of a three fields model for a quasi-Newtonian flow, Comput. Methods Appl. Mech. Engrg., 109, 281-292 (1993) · Zbl 0844.76004 |
| [6] | Barrett, J. B.; Liu, W. B., Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math., 68, 437-456 (1994) · Zbl 0811.76036 |
| [7] | Bird, R. B.; Armstrong, R. C.; Hassager, O., Dynamics of Polymeric Liquids I (1987), John Wiley and Sons: John Wiley and Sons Amsterdam |
| [8] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [9] | Amri, H. El, (Thèse d’Etat (1990), Université Lyon I) |
| [10] | Girault, V.; Raviart, P. A., Finite element method for Navier-Stokes equations, (Theory and Algorithms (1986), Springer: Springer Berlin-Heidelberg-New York) · Zbl 0396.65070 |
| [11] | Loula, A. F.D.; Guerreiro, J. N.C., A new finite element method for nonlinear creeping flows, Comput. Methods Appl. Mech. Engrg., 79, 1, 87-109 (1990) · Zbl 0716.73091 |
| [12] | Sandri, D., Sur l’approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau, \(M^2\) AN, 27, 131-155 (1991) · Zbl 0764.76039 |
| [13] | Verfürth, R., A Review of a posteriori Error Estimation and Adaptative Mesh-Refinement Techniques (1996), Wiley Teubner: Wiley Teubner Chichester · Zbl 0853.65108 |
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