PEERS: A new mixed finite element for plane elasticity. (English) Zbl 0633.73074

A mixed finite element procedure for plane elasticity is introduced and analyzed. The symmetry of the stress tensor is enforced through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is introduced to simplify the linear algebraic system. Applications are made to incompressible elastic problems and to plasticity problems.


74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65K10 Numerical optimization and variational techniques
Full Text: DOI


[1] M. Amara and J. M. Thomas, Equilibrium finite elements for the linear elastic problem. Numer. Math.,33 (1979), 367–383. · Zbl 0401.73079
[2] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing, and error estimates. RAIRO Anal. Numér.,19 (1985). · Zbl 0567.65078
[3] D. N. Arnold, J. Douglas, Jr., and C. P. Gupta, A family of higher order finite element methods for plane elasticity. Numer. Math.,45 (1984), 1–22. · Zbl 0558.73066
[4] D. N. Arnold, L. R. Scott, and M. Vogelius, Regular solutions of divu=f with Dirichlet boundary conditions on a polygon. Tech. Note, Univ. Maryland, to appear. · Zbl 0702.35208
[5] F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér.,2 (1974), 129–151. · Zbl 0338.90047
[6] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.,9 (1975), 33–76.
[7] J. Douglas, Jr., and F. A. Milner, Interior and superconvergence estimates for mixed methods for second order elliptic problems, to appear in RAIRO Anal. Numér.
[8] J. Douglas, Jr., and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Aplic. Comp.,1 (1982), 91–103. · Zbl 0482.65057
[9] J. Douglas, Jr., and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, to appear in Math. Comput.
[10] I. Ekeland and R. Temam,Analyse Convexe et Problèmes Variationnels. Dunod-Gauthier-Villars, Paris, 1974. · Zbl 0281.49001
[11] R. S. Falk and J. E. Osborn, Error estimates for mixed methods. RAIRO Anal. Numér.,14 (1980), 309–324.
[12] M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér.,11 (1977), 341–354. · Zbl 0373.65055
[13] B. X. Fraeijs de Veubeke, Stress function approach. World Congress on the Finite Element Method in Structural Mechanics, Bornemouth, 1975.
[14] C. Johnson, On finite element methods for plasticity problems. Numer. Math.,26 (1976), 79–84. · Zbl 0355.73035
[15] C. Johnson, A mixed finite element method for plasticity with hardening. SIAM J. Numer. Anal.,14 (1977), 575–583. · Zbl 0374.73039
[16] C. Johnson, Existence theorems for plasticity problems. J. Math. Pure Appl.,55 (1976), 431–444. · Zbl 0351.73049
[17] C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math.,30 (1978), 103–116. · Zbl 0427.73072
[18] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems.Mathematical Aspects of the Finite Element Method (eds., I. Galligani and E. Magenes), Lecture Notes in Math. 606, Springer-Verlag, 1977.
[19] M. Vogelius, An analysis of thep-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal order estimates. Numer. Math.,41 (1983), 39–53. · Zbl 0504.65061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.