The Mortar finite element method for Bingham fluids
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 1, p. 153-164

This paper deals with the flow problem of a viscous plastic fluid in a cylindrical pipe. In order to approximate 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.

On considère le problème de l'écoulement d'un fluide visqueux plastique dans une conduite cylindrique. Afin d'approcher ce problème régi par une inéquation variationnelle, nous appliquons la méthode non conforme des éléments finis avec joints. En utilisant des techniques appropriées, on devient en mesure de prouver la convergence de la méthode avec un taux de convergence identique au cas conforme.

Classification:  65N30,  65N55,  76A05
Keywords: viscoplastic fluid, Bingham model, variational inequality, mortar finite element method, a priori error estimates
@article{M2AN_2001__35_1_153_0,
     author = {Hild, Patrick},
     title = {The Mortar finite element method for Bingham fluids},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {1},
     year = {2001},
     pages = {153-164},
     zbl = {0990.76042},
     mrnumber = {1811985},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_1_153_0}
}
Hild, Patrick. The Mortar finite element method for Bingham fluids. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 1, pp. 153-164. http://www.numdam.org/item/M2AN_2001__35_1_153_0/

[1] Y. Achdou and O. Pironneau, 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. | Zbl 0833.76032

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

[3] F. Ben Belgacem, The mixed mortar finite element method for the incompressible Stokes problem: Convergence analysis. SIAM J. Numer. Anal. 37 (2000) 1085-1100. | Zbl 0959.65126

[4] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. | Zbl 0944.65114

[5] F. Ben Belgacem, P. Hild and P. Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 9 (1999) 287-303. | Zbl 0940.74056

[6] C. Bernardi, N. Debit and Y. Maday, Coupling finite element and spectral methods: First results. Math. Comp. 54 (1990) 21-39. | Zbl 0685.65098

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

[8] 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. | Zbl 0797.65094

[9] P.E. Bjørstad and O.B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1097-1120. | Zbl 0615.65113

[10] 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. | MR 394323 | Zbl 0278.47033

[11] 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. | Zbl 0875.65086

[12] 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).

[13] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | Zbl 0298.73001

[14] R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28 (1974) 963-971. | Zbl 0297.65061

[15] R. Glowinski, Lectures on numerical methods for non-linear variational problems. Springer, Berlin (1980). | Zbl 0456.65035

[16] P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles. Ph.D. thesis, University of Toulouse III, France (1998).

[17] J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure. Appl. Math. XX (1967) 493-519. | Zbl 0152.34601

[18] 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. | Zbl 0168.45505

[19] 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. | Zbl 0168.45601

[20] 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. | Zbl 0236.76006