Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Analysis of some mixed finite element methods for plane elasticity equations

Authors: J. Pitkäranta and R. Stenberg
Journal: Math. Comp. 41 (1983), 399-423
MSC: Primary 65N15; Secondary 73K25
MathSciNet review: 717693
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze some mixed finite element methods, based on rectangular elements, for solving the two-dimensional elasticity equations. We prove error estimates for a method proposed by Taylor and Zienkiewicz and for some new variants of the known equilibrium methods. A numerical example is given demonstrating the performance of the various algorithms considered.

References [Enhancements On Off] (What's this?)

  • [1] Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322–333. MR 0288971
  • [2] I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039–1062. MR 583486, 10.1090/S0025-5718-1980-0583486-7
  • [3] M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér. 12 (1978), no. 3, 211–236, iii (English, with French summary). MR 509973
  • [4] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 0263214
  • [5] F. 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 (English, with loose French summary). MR 0365287
  • [6] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [7] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
  • [8] I. Hlavaček, "Convergence of an equilibrium finite element method for plane elastostatics," Apt. Math., v. 24, 1979, pp. 427-456.
  • [9] C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103–116. MR 0483904
  • [10] Claes Johnson and Juhani Pitkäranta, Analysis of some mixed finite element methods related to reduced integration, Math. Comp. 38 (1982), no. 158, 375–400. MR 645657, 10.1090/S0025-5718-1982-0645657-2
  • [11] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, Mass., 1973.
  • [12] D. Malkus & T. Hughes, "Mixed finite element methods--reduced and selective integration techniques: A unification of concepts," Comp. Methods Appl. Mech. Engrg., v. 15, 1978, pp. 63-81.
  • [13] J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237–248 (English, with French summary). MR 631678
  • [14] G. Sander, Application of the Dual Analysis Principle, Proc. of IUTAM Symp. on High Speed Computing of Elastic Structures, Univ. de Liège, 1971, pp. 167-207.
  • [15] R. L. Taylor and O. C. Zienkiewicz, Complementary energy with penalty functions in finite element analysis, Energy methods in finite element analysis, Wiley, Chichester, 1979, pp. 153–174. MR 537004
  • [16] V. B. Watwood, Jr. & B. J. Hartz, "An equilibrium stress field model for finite element solutions of two-dimensional elastostatic problems," Internat. J. Solids and Structures, v. 4, 1968, pp. 857-873.
  • [17] B. Fraejis de Veubeke, "Displacement and equilibrium models in the finite element method," in Stress Analysis (O. C. Zienkiewicz and G. S. Holister, eds.), Wiley, New York, 1965, pp. 145-197.
  • [18] B. Fraejis de Veubeke, Finite Element Method in Aerospace Engineering Problems, Proc. Internat. Sympos. Computing Methods in Appl. Sci. and Eng., Versailles, 1973, Part 1, pp. 224-258.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N15, 73K25

Retrieve articles in all journals with MSC: 65N15, 73K25

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society