Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-13T11:39:44.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-element discretizations of a two-dimensional grade-two fluidmodel

Published online by Cambridge University Press:  15 April 2002

Vivette Girault and
Larkin Ridgway Scott
Vivette Girault
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
Larkin Ridgway Scott
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1581, USA.
Get access

Abstract

We propose and analyze several finite-element schemes for solving a grade-twofluid model, with atangential boundary condition, in a two-dimensional polygon. The exactproblem is split into ageneralized Stokes problem and a transport equation, in such a way that italways has a solutionwithout restriction on the shape of the domain and on the size of the data.The first scheme usesdivergence-free discrete velocities and a centered discretization of thetransport term, whereas theother schemes use Hood-Taylor discretizations for the velocity andpressure, and either a centered or an upwinddiscretization of the transport term. One facet of our analysis isthat, without restrictions on the data,each scheme has a discrete solution and all discrete solutions convergestrongly to solutions of theexact problem. Furthermore, if the domain is convex and the data satisfycertain conditions, eachscheme satisfies error inequalities that lead to error estimates.

Keywords

Mixed formulationdivergence-zerofinite elementsinf-sup conditionuniform W1,p -stabilityHood-Taylor methodstreamline diffusion.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001, pp. 1007 - 1053
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
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.
Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. CrossRef
Baia, M. and Sequeira, A., A finite element approximation for the steady solution of a second-grade fluid model. J. Comput. Appl. Math. 111 (1999) 281-295.
Bernardi, C. and Girault, V., A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. CrossRef
Boland, J. and Nicolaides, R., Stabilility of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. CrossRef
S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Texts in Applied Mathematics 15 , Springer-Verlag, New York (1994).
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151.
Brezzi, F. and Falk, R.S., Stability of a higher order Hood-Taylor method. SIAM J. Numer. Anal. 28 (1991) 581-590. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. (1975) 77-84.
Cioranescu, D. and Ouazar, E.H., Existence et unicité pour les fluides de second grade. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 285-287.
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.
Dunn, J.E. and Fosdick, R.L., Thermodynamics, stability, and boundedness of fluids of complexity two and fluids of second grade. Arch. Rational Mech. Anal. 56 (1974) 191-252. CrossRef
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.
Durán, R., Nochetto, R.H. and Wang, J., Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D. Math. Comp. 51 (1988) 1177-1192. CrossRef
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).
Girault, V. and Scott, L.R., Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition. J. Math. Pures Appl. 78 (1999) 981-1011. CrossRef
V. Girault and L.R. Scott, Hermite Interpolation of Non-Smooth Functions Preserving Boundary Conditions. Department of Mathematics, University of Chicago, Preprint (1999).
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.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24 Pitman, Boston (1985).
Holm, D.D., Marsden, J.E. and Ratiu, T.S., Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349 (1998) 4173-4177. CrossRef
Holm, D.D., Marsden, J.E. and Ratiu, T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. in Math. 137 (1998) 1-81. CrossRef
Hopf, E., Über die Aufangswertaufgabe für die hydrodynamischen Grundleichungen. Math. Nachr. 4 (1951) 213-231. CrossRef
Hugues, T.J.R., A simple finite element scheme for developping upwind finite elements. Internat. J. Numer. Methods Engrg. 12 (1978) 1359-1365. CrossRef
C. Johnson, Numerical Solution of PDE by the Finite Element Method. Cambridge University Press, Cambridge (1987).
Johnson, C., Nävert, U. and Pitkäranta, J., Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1985) 285-312. CrossRef
Leray, J., Étude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933) 1-82.
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I. Dunod, Paris (1968).
Morgan, J.W. and Scott, L.R., A nodal basis for C 1 piecewise polynomials of degree n ≥ 5. Math. Comp. 29 (1975) 736-740.
J. Necas, Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
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).
Peetre, J., Espaces d'interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16 (1966) 279-317. CrossRef
O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989).
Scott, L.R. and Vogelius, M., 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. CrossRef
Scott, L.R. and Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. CrossRef
Stenberg, R., Analysis of finite element methods for the Stokes problem: a unified approach. Math. Comp. 42 (1984) 9-23.
L. Tartar, Topics in nonlinear analysis, in Publications Mathématiques d'Orsay, Université Paris-Sud, Orsay (1978).