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.

**[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.

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717693-X

Article copyright:
© Copyright 1983
American Mathematical Society