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

Girault, Vivette; Scott, Larkin Ridgway

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1007-1053.

We propose and analyze several finite-element schemes for solving a grade-two fluid model, with a tangential boundary condition, in a two-dimensional polygon. The exact problem is split into a generalized Stokes problem and a transport equation, in such a way that it always has a solution without restriction on the shape of the domain and on the size of the data. The first scheme uses divergence-free discrete velocities and a centered discretization of the transport term, whereas the other schemes use Hood-Taylor discretizations for the velocity and pressure, and either a centered or an upwind discretization of the transport term. One facet of our analysis is that, without restrictions on the data, each scheme has a discrete solution and all discrete solutions converge strongly to solutions of the exact problem. Furthermore, if the domain is convex and the data satisfy certain conditions, each scheme satisfies error inequalities that lead to error estimates.

Classification : 65D30, 65N15, 65N30
Mots clés : mixed formulation, divergence-zero finite elements, inf-sup condition, uniform $W^{1,p}$-stability, Hood-Taylor method, streamline diffusion

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] M. Amara, C. Bernardi and V. Girault, Conforming and nonconforming discretizations of a two-dimensional grade-two fluid. In preparation.

[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. | Numdam | Zbl

[4] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | Zbl

[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. | Zbl

[6] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. | Zbl

[7] J. Boland and R. Nicolaides, Stabilility of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | Zbl

[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). | MR | Zbl

[9] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151. | Numdam | Zbl

[10] F. Brezzi and R.S. Falk, Stability of a higher order Hood-Taylor method. SIAM J. Numer. Anal. 28 (1991) 581-590. | Zbl

[11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl

[12] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR | Zbl

[13] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. (1975) 77-84. | Numdam | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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). | MR | Zbl

[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. | Zbl

[21] V. Girault and L.R. Scott, Hermite Interpolation of Non-Smooth Functions Preserving Boundary Conditions. Department of Mathematics, University of Chicago, Preprint (1999). | Zbl

[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. | MR | Zbl

[23] P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24 Pitman, Boston (1985). | MR | Zbl

[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] 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. | Zbl

[26] E. Hopf, Über die Aufangswertaufgabe für die hydrodynamischen Grundleichungen. Math. Nachr. 4 (1951) 213-231. | Zbl

[27] T.J.R. Hugues, A simple finite element scheme for developping upwind finite elements. Internat. J. Numer. Methods Engrg. 12 (1978) 1359-1365. | Zbl

[28] C. Johnson, Numerical Solution of PDE by the Finite Element Method. Cambridge University Press, Cambridge (1987). | MR | Zbl

[29] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1985) 285-312. | Zbl

[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. | Numdam | Zbl

[31] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | MR | Zbl

[32] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I. Dunod, Paris (1968). | Zbl

[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. | Zbl

[34] J. Nečas, Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR

[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] E.H. Ouazar, Sur les fluides de second grade. Thèse de 3ème Cycle, Université Paris VI (1981).

[37] J. Peetre, Espaces d'interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16 (1966) 279-317. | Numdam | Zbl

[38] O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989). | MR | Zbl

[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. | Numdam | Zbl

[40] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl

[41] R. Stenberg, Analysis of finite element methods for the Stokes problem: a unified approach. Math. Comp. 42 (1984) 9-23. | Zbl

[42] L. Tartar, Topics in nonlinear analysis, in Publications Mathématiques d'Orsay, Université Paris-Sud, Orsay (1978). | Zbl