Finite-element discretizations of a two-dimensional grade-two fluid model

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1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
2. [2] M. Amara, C. Bernardi and V. Girault, Conforming and nonconforming discretizations of a two-dimensional grade-two fluid. In preparation. Zbl1072.76036
3. [3] D.N. Arnold, L.R. Scott and M. Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Ser. 15 (1988) 169–192. Zbl0702.35208
4. [4] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179–192. Zbl0258.65108
5. [5] M. Baia and A. Sequeira, A finite element approximation for the steady solution of a second-grade fluid model. J. Comput. Appl. Math. 111 (1999) 281–295. Zbl0957.76033
6. [6] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893–1916. Zbl0913.65007
7. [7] J. Boland and R. Nicolaides, Stabilility of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722–731. Zbl0521.76027
8. [8] S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Texts in Applied Mathematics 15, Springer-Verlag, New York (1994). Zbl0804.65101MR1278258
9. [9] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129–151. Zbl0338.90047
10. [10] F. Brezzi and R.S. Falk, Stability of a higher order Hood-Taylor method. SIAM J. Numer. Anal. 28 (1991) 581–590. Zbl0731.76042
11. [11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
12. [12] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). Zbl0383.65058MR520174
13. [13] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. (1975) 77–84. Zbl0368.65008
14. [14] D. Cioranescu and E.H. Ouazar, Existence et unicité pour les fluides de second grade. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 285–287. Zbl0571.76005
15. [15] D. Cioranescu and E.H. Ouazar, Existence and uniqueness for fluids of second grade, in Nonlinear Partial Differential Equations, Collège de France Seminar 109, Pitman (1984) 178–197. Zbl0577.76012
16. [16] J.E. Dunn and R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity two and fluids of second grade. Arch. Rational Mech. Anal. 56 (1974) 191–252. Zbl0324.76001
17. [17] J.E. Dunn and K.R. Rajagopal, Fluids of differential type: Critical review and thermodynamic analysis. Internat. J. Engrg. Sci. 33 5 (1995) 689–729. Zbl0899.76062
18. [18] R. Durán, R.H. Nochetto and J. Wang, Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-$D$. Math. Comp. 51 (1988) 1177–1192. Zbl0699.76038
19. [19] V. Girault and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms, in Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin (1986). Zbl0585.65077MR851383
20. [20] V. Girault and L.R. Scott, Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition. J. Math. Pures Appl. 78 (1999) 981–1011. Zbl0961.35116
21. [21] V. Girault and L.R. Scott, Hermite Interpolation of Non-Smooth Functions Preserving Boundary Conditions. Department of Mathematics, University of Chicago, Preprint (1999). Zbl1002.65129
22. [22] V. Girault and L.R. Scott, An upwind discretization of a steady grade-two fluid model in two dimensions. To appear in Collège de France Seminar. Zbl1034.35110MR1936003
23. [23] P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24 Pitman, Boston (1985). Zbl0695.35060MR775683
24. [24] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349 (1998) 4173–4177. 
25. [25] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. in Math. 137 (1998) 1–81. Zbl0951.37020
26. [26] E. Hopf, Über die Aufangswertaufgabe für die hydrodynamischen Grundleichungen. Math. Nachr. 4 (1951) 213–231. Zbl0042.10604
27. [27] T.J.R. Hugues, A simple finite element scheme for developping upwind finite elements. Internat. J. Numer. Methods Engrg. 12 (1978) 1359–1365. Zbl0393.65044
28. [28] C. Johnson, Numerical Solution of PDE by the Finite Element Method. Cambridge University Press, Cambridge (1987). Zbl0628.65098MR925005
29. [29] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1985) 285–312. Zbl0526.76087
30. [30] J. Leray, Étude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12 (1933) 1–82. Zbl0006.16702
31. [31] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). Zbl0189.40603MR259693
32. [32] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I. Dunod, Paris (1968). Zbl0165.10801
33. [33] J.W. Morgan and L.R. Scott, A nodal basis for $C^1$ piecewise polynomials of degree $n\ge 5$. Math. Comp. 29 (1975) 736–740. Zbl0307.65074
34. [34] J. Nečas, Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). MR227584
35. [35] R.R. Ortega, Contribución al estudio teórico de algunas E.D.P. no lineales relacionadas con fluidos no Newtonianos. Thesis, University of Sevilla (1995). 
36. [36] E.H. Ouazar, Sur les fluides de second grade. Thèse de 3ème Cycle, Université Paris VI (1981). 
37. [37] J. Peetre, Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16 (1966) 279–317. Zbl0151.17903
38. [38] O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989). Zbl0712.76001MR1030279
39. [39] L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111–143. Zbl0608.65013
40. [40] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. Zbl0696.65007
41. [41] R. Stenberg, Analysis of finite element methods for the Stokes problem: a unified approach. Math. Comp. 42 (1984) 9–23. Zbl0535.76037
42. [42] L. Tartar, Topics in nonlinear analysis, in Publications Mathématiques d’Orsay, Université Paris-Sud, Orsay (1978). Zbl0395.00008