Finite-element discretizations of a two-dimensional grade-two fluid model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, p. 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
Keywords: mixed formulation, divergence-zero finite elements, inf-sup condition, uniform ${W}^{1,p}$-stability, Hood-Taylor method, streamline diffusion
@article{M2AN_2001__35_6_1007_0,
author = {Girault, Vivette and Scott, Larkin Ridgway},
title = {Finite-element discretizations of a two-dimensional grade-two fluid model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
pages = {1007-1053},
zbl = {1032.76033},
mrnumber = {1873516},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_6_1007_0}
}

Girault, Vivette; Scott, Larkin Ridgway. Finite-element discretizations of a two-dimensional grade-two fluid model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, pp. 1007-1053. http://www.numdam.org/item/M2AN_2001__35_6_1007_0/

 R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

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

 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 0702.35208

 I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | Zbl 0258.65108

 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 0957.76033

 C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. | Zbl 0913.65007

 J. Boland and R. Nicolaides, Stabilility of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | Zbl 0521.76027

 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 1278258 | Zbl 0804.65101

 F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151. | Numdam | Zbl 0338.90047

 F. Brezzi and R.S. Falk, Stability of a higher order Hood-Taylor method. SIAM J. Numer. Anal. 28 (1991) 581-590. | Zbl 0731.76042

 F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

 P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

 P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. (1975) 77-84. | Numdam | Zbl 0368.65008

 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 0571.76005

 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 0577.76012

 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 0324.76001

 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 0899.76062

 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 0699.76038

 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 851383 | Zbl 0585.65077

 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 0961.35116

 V. Girault and L.R. Scott, Hermite Interpolation of Non-Smooth Functions Preserving Boundary Conditions. Department of Mathematics, University of Chicago, Preprint (1999). | Zbl 1002.65129

 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 1936003 | Zbl 1034.35110

 P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24 Pitman, Boston (1985). | MR 775683 | Zbl 0695.35060

 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.

 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 0951.37020

 E. Hopf, Über die Aufangswertaufgabe für die hydrodynamischen Grundleichungen. Math. Nachr. 4 (1951) 213-231. | Zbl 0042.10604

 T.J.R. Hugues, A simple finite element scheme for developping upwind finite elements. Internat. J. Numer. Methods Engrg. 12 (1978) 1359-1365. | Zbl 0393.65044

 C. Johnson, Numerical Solution of PDE by the Finite Element Method. Cambridge University Press, Cambridge (1987). | MR 925005 | Zbl 0628.65098

 C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1985) 285-312. | Zbl 0526.76087

 J. Leray, Étude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933) 1-82. | Zbl 0006.16702

 J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | MR 259693 | Zbl 0189.40603

 J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I. Dunod, Paris (1968). | Zbl 0165.10801

 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 0307.65074

 J. Nečas, Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR 227584

 R.R. Ortega, Contribución al estudio teórico de algunas E.D.P. no lineales relacionadas con fluidos no Newtonianos. Thesis, University of Sevilla (1995).

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

 J. Peetre, Espaces d'interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16 (1966) 279-317. | Numdam | Zbl 0151.17903

 O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989). | MR 1030279 | Zbl 0712.76001

 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 0608.65013

 L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl 0696.65007

 R. Stenberg, Analysis of finite element methods for the Stokes problem: a unified approach. Math. Comp. 42 (1984) 9-23. | Zbl 0535.76037

 L. Tartar, Topics in nonlinear analysis, in Publications Mathématiques d'Orsay, Université Paris-Sud, Orsay (1978). | Zbl 0395.00008