Analysis of a three-field formulation of the Stokes problem. (Analyse d’une formulation à trois champs du problème de Stokes.) (French) Zbl 0791.76008

Summary: We study the numerical approximation of a three-fields formulation of Stokes problem. The unknowns are extra stress tensor, velocity and pressure. This formulation is motivated by the study of viscoelastic fluids obeying Oldroyd constitutive equation. We study the inf-sup conditions relating extra stress tensor, velocity and pressure. We give some examples of finite element spaces satisfying or not satisfying these conditions and we conclude by the study of a fixed point method to solve the approximate problem.


76A10 Viscoelastic fluids
35Q30 Navier-Stokes equations
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI EuDML


[1] [1] I. BABUSKA, 1971, Error-bounds for finite element method, Numer. Math., 16, 322-333. Zbl0214.42001 MR288971 · Zbl 0214.42001
[2] [2] J. BARANGER, D. SANDRI, 1992, Formulation of Stokes’s problem and the linear elasticity equations suggested by Oldroyd model for viscoelastics flows, RAIRO ModéL Math. Anal Numér., 26, 331-345. Zbl0738.76002 MR1153005 · Zbl 0738.76002
[3] [3] F. BREZZI, 1974, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Modél. Math. Anal. Numér., 8, 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047
[4] P. G. CIARLET, 1978, The finite element method for elliptic problems, North-Holland, Amsterdam. Zbl0383.65058 MR520174 · Zbl 0383.65058
[5] M. FORTIN, R. PIERRE, 1989, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows, Comput. Methods Appl. Mech. Engrg., 73, 341-350. Zbl0692.76002 MR1016647 · Zbl 0692.76002
[6] V. GIRAULT, P. A. RAVIART, 1986, Finite element method for Navier-Stokes equations, Theory and Algorithms, Springer, Berlin Heidelberg New York. Zbl0585.65077 MR851383 · Zbl 0585.65077
[7] R. KEUNINGS, 1989, in : Tucker Ch. III (éd.), Computer Modeling for Polymer Processing, 403-469. Munich: Hanser Verlag.
[8] J. M. MARCHAL, M. J. CROCHET, 1987, A new finite element for calculating viscoelastic flow, J. Non-Newtonian Fluid Mech., 26, 77-114. Zbl0637.76009 · Zbl 0637.76009
[9] V. RUAS, An optimal three field finite element approximation of the Stokes system with continuous extra stresses, Japan Journal of Industrial and Applied Mathematics, à paraître. Zbl0797.76045 · Zbl 0797.76045
[10] K. YOSHIDA, 1980, Functional Analysis, Springer Verlag, Berlin Heidelberg New York.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.