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A formulation of Stokes’ problem and the linear elasticity equations suggested by the Oldroyd model for viscoelastic flow. (English) Zbl 0738.76002
Summary: We propose a three fields formulation of Stokes’s problem and the equations of linear elasticity, allowing confirming finite element approximation and using only the classical inf-sup condition relating velocity and pressure. No condition of this type is needed on the “non Newtonian” extra stress tensor. For the linear elasticity equations this method gives uniform results with respect to the compressibility.

MSC:
76A10 Viscoelastic fluids
76D07 Stokes and related (Oseen, etc.) flows
74B05 Classical linear elasticity
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[1] [1] D. N. ARNOLD, J. DOUGLAS and C. P. GUPTA, A Family of Higher Order Mixed Finite Element Methods for Plane Elasticity, Numer. Math., 45, 1-22 (1984). Zbl0558.73066 MR761879 · Zbl 0558.73066
[2] [2] I. BABUSKA, Error-bounds for Finite Element Method, Numer. Math., 16, 322-333 (1971). Zbl0214.42001 MR288971 · Zbl 0214.42001
[3] [3] F. BREZZI, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Model. Math. Anal. Numér., 8, 129-151 (1974). Zbl0338.90047 MR365287 · Zbl 0338.90047
[4] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland (1978). Zbl0383.65058 MR520174 · Zbl 0383.65058
[5] [5] P. CLEMENT, Approximation by finite elements using local regularization, RAIRO Modél. Math. Anal. Numér., 8, 77-84 (1975). Zbl0368.65008 MR400739 · Zbl 0368.65008
[6] J. DOUGLAS and J. WANG, An absolutely stabilized finite element method for the Stokes problem, quoted in [12]. Zbl0669.76051 · Zbl 0669.76051
[7] M. FORTIN and A. FORTIN, A new approach for the FEM simulation of viscoelastic flows, J. Non-Newtonian Fluid Mech., 32, 295-310 (1989). Zbl0672.76010 · Zbl 0672.76010
[8] M. FORTIN and R. PIERRE, On the convergence of the mixed method of Crochetand Marchal for viscoelastic flows, to appear. Zbl0692.76002 MR1016647 · Zbl 0692.76002
[9] L. P. FRANCA, Analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle, Comp. Meth. Appl. Mech. Engrg., 76, 259-273 (1989). Zbl0688.73044 MR1030385 · Zbl 0688.73044
[10] L. P. FRANCA and T. J. R. HUGHES, Two classes of mixed finite element methods, Comp. Meth. Appl. Mech. Engrg., 69, 89-129 (1988). Zbl0629.73053 MR953593 · Zbl 0629.73053
[11] L. P. FRANCA, R. STENBERG, Finite element approximation of a new variational principle for compressible and incompressible linear isotropic elasticity, to appear in Appl. Mech. Rev. Zbl0749.73076 MR1037580 · Zbl 0749.73076
[12] L.P. FRANCA and R. STENBERG, Error analysis of some Galerkin-least-squares methods for the elasticity equations, Rapport INRIA, n^\circ 1054 (1989). Zbl0759.73055 · Zbl 0759.73055
[13] V. GIRAULT and P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Theory and algorithms, Springer Berlin (1978). Zbl0585.65077 MR851383 · Zbl 0585.65077
[14] J. M. MARCHAL and M. J. CROCHET, A new mixed finite element for calculating viscoelastic flow, J. Non-Newtonian Fluid Mech., 26, 77-114 (1987). Zbl0637.76009 · Zbl 0637.76009
[15] [15] L. R. SCOTT and M. VOGELIUS, Norm estimates for a maximal right inverse ofthe divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér., 19, 111-143 (1985). Zbl0608.65013 MR813691 · Zbl 0608.65013
[16] [16] R. STENBERG, A Family of Mixed Finite Elements for the Elasticity Problem, Num. Math., 53, 513-538 (1988). Zbl0632.73063 MR954768 · Zbl 0632.73063
[17] R. STENBERG, Error Analysis of some Finite Element Methods for the Stokes Problem, to appear. Zbl0702.65095 MR1010601 · Zbl 0702.65095
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