Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T21:30:40.777Z Has data issue: false hasContentIssue false
  • English
  • Français

The mortar finite element method for Bingham fluids

Published online by Cambridge University Press:  15 April 2002

Patrick Hild
Patrick Hild*
Affiliation:
Laboratoire de Mathématiques, Université de Savoie, CNRS EP 2067, 73376 Le Bourget-du-Lac, France. (hild@univ-savoie.fr)
Get access

Abstract

This paper deals with the flow problem of aviscous plastic fluid in a cylindrical pipe. In order toapproximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.

Keywords

Viscoplastic fluidBingham modelvariational inequalitymortar finite element methoda priori error estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 1 , January 2001 , pp. 153 - 164
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.)

References

Achdou, Y. and Pironneau, O., A fast solver for Navier-Stokes equations in the laminar regime using mortar finite elements and boundary element methods. SIAM J. Numer. Anal. 32 (1995) 985-1016. CrossRef
R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Ben Belgacem, F., The mixed mortar finite element method for the incompressible Stokes problem: Convergence analysis. SIAM J. Numer. Anal. 37 (2000) 1085-1100. CrossRef
Ben Belgacem, F., The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197.
Ben Belgacem, F., Hild, P. and Laborde, P., Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 9 (1999) 287-303.
Bernardi, C., Debit, N. and Maday, Y., Coupling finite element and spectral methods: First results. Math. Comp. 54 (1990) 21-39. CrossRef
Bernardi, C. and Girault, V., Local, A regularisation operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. CrossRef
C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Collège de France Seminar, H. Brezis and J.-L. Lions Eds., Pitman (1994) 13-51.
Bjrstad, P.E. and Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1097-1120.
H. Brezis, Monotonicity in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to nonlinear functional analysis, E. Zarantonello Ed., Academic Press, New York (1971) 101-156.
P.-G. Ciarlet, The finite element method for elliptic problems, in Handbook of numerical analysis, Vol. II, Part 1, P.-G. Ciarlet and J.-L. Lions Eds., North Holland, Amsterdam (1991) 17-352.
N. Debit, La méthode des éléments avec joints dans le cas du couplage de méthodes spectrales et méthodes d'éléments finis: résolution des équations de Navier-Stokes. Ph.D. thesis, University of Paris VI, France (1991).
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
Falk, R.S., Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28 (1974) 963-971. CrossRef
R. Glowinski, Lectures on numerical methods for non-linear variational problems. Springer, Berlin (1980).
P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles. Ph.D. thesis, University of Toulouse III, France (1998).
J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure. Appl. Math. XX (1967) 493-519.
P.P. Mosolov and V.P. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium. PPM, J. Mech. Appl. Math. 29 (1965) 545-577.
P.P. Mosolov and V.P. Miasnikov, On stagnant flow regions of a viscous-plastic medium in pipes. PPM, J. Mech. Appl. Math. 30 (1966) 841-854.
P.P. Mosolov and V.P. Miasnikov, On qualitative singularities of the flow of a viscoplastic medium in pipes. PPM, J. Mech Appl. Math. 31 (1967) 609-613.