Abstract
A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed in [8] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case.
The work of the first author was supported in part by NSF grant DMS-0411388.
The work of the second author was supported in part by NSF grant DMS03-08347.
The work of the third author was supported by the Norwegian Research Council.
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References
Mohamed Amara and Jean-Marie Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979), no. 4, 367–383.
Douglas N. Arnold, Franco Brezzi, and Jim Douglas, Jr., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347–367.
Douglas N. Arnold, Jim Douglas, Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22.
Douglas N. Arnold and Richard S. Falk, A new mixed formulation for elasticity, Numer. Math. 53 (1988), no. 1–2, 13–30.
Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. on Numer. Anal. 26 (1989), 1276–1290.
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. I. The de Rham complex, this volume.
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, preprint.
Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), no. 3, 401–419.
Christine Bernardi and Geneviéve Raugel, Analysis of some finite elements for the Stokes problem, Math. Comp. 44 (1985), 71–79.
I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, Differential operators on the baseaffine space and a study of gmodules, Lie groups and their representation, I.M. Gelfand (ed) (1975), 21–64.
Franco Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151.
Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
Michael Eastwood, A complex from linear elasticity, Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 23–29.
Michel Fortin, Old and new finite elements for incompressible flows, Int. J. Numer. Methods Fluids 1 (1981), 347–354.
Baudoiun M. Fraejis de Veubeke, Stress function approach, World Congress on the Finite Element Method in Structural Mechanics, Bornemouth, 1975.
V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986.
Claes Johnson and Bertrand Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103–116.
Mary E. Morley, A family of mixed finite elements for linear elasticity, Numer. Math. 55 (1989), no. 6, 633–666.
Erwin Stein and Raimund Rolfes, Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity, Comput. Methods Appl. Mech. Engrg. 84 (1990), no. 1, 77–95.
Rolf Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. Math. 48 (1986), no. 4, 447–462.
—, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538.
—, Two low-order mixed methods for the elasticity problem, The mathematics of finite elements and applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 271–280.
Vernon B. Watwood Jr. and B.J. Hartz, An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems, Internat. Jour. Solids and Structures 4 (1968), 857–873.
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Arnold, D.N., Falk, R.S., Winther, R. (2006). Defferential Complexes and Stability of Finite Element Methods II: The Elasticity Complex. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_3
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DOI: https://doi.org/10.1007/0-387-38034-5_3
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