Tamtam Proceedings - lamsin

272 El-Kyal et al.Theorem 3.1 There exist positive constants M 0 , h 0 and k 0 such that if problem (O)admits a solution (σ, u, p) ∈ W 2,∞ (Ω) × ( W 3,2 (Ω) ∩ W 2,∞ (Ω) ) × ( H 2 (Ω) ∩ L 2 0(Ω) )satisfying}max{‖ σ ‖ 2,∞ , ‖ u ‖ 3,2 , ‖ u ‖ 2,∞ , ‖ p ‖ 2,2 ≤ M 0 ,√then for all h ≤ h 0 , k ≤ k 0 such that ∃ a 1 , a 2 : a 1 k ≤ h ≤ a 2 k, problem (Okh) admits asolution (σh k, uk h , pk h ) ∈ T h × X h × Q h and there exists a positive constant C independentof h and k such that( ) h| σ − σh k | + | d(u − u k 2h) | ≤ C √ + k , (8)k( ) h| p − p k 2h | ≤ C √ + k . (9)k4. References[1] N. PHAN-THIEN, RI. TANNER. A new constitutive equation derived from network theory, J.Non-Newtonian Fluid Mech. 2 (1977) 353-365.[2] J. BARANGER, A. MACHMOUM. A "natural" norm for the method of characteristics, to appearin M 2 AN.[3] A. MACHMOUM, D. ESSELAOUI. Finite element approximation of viscoelastic fluid flow usingcharacteristics method, Comput. Methods Appl. Mech. Engrg. 190 (2001) 5603-5618.[4] J. BARANGER, D. SANDRI. Finite element approximation of viscoelastic fluid flow Existenceof solutions and error bounds. I-Discontinuous constraints, Numer. Math. 63 (1992) 13-27.[5] V. GIRAULT, P. A. RAVIART. Finite element method for Navier Stokes equations, Theory andAlgorithms. Berlin Heidelberg New York : Springer 1986.TAMTAM –Tunis– 2005