Analysis of some mixed finite element methods for plane elasticity equations
Authors:
J. Pitkäranta and R. Stenberg
Journal:
Math. Comp. 41 (1983), 399423
MSC:
Primary 65N15; Secondary 73K25
MathSciNet review:
717693
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Abstract: We analyze some mixed finite element methods, based on rectangular elements, for solving the twodimensional 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, Errorbounds for finite element method, Numer.
Math. 16 (1970/1971), 322–333. MR 0288971
(44 #6166)
 [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
(81m:65166), http://dx.doi.org/10.1090/S00255718198005834867
 [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
(80b:49031)
 [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
(41 #7819)
 [5]
F.
Brezzi, On the existence, uniqueness and approximation of
saddlepoint problems arising from Lagrangian multipliers, Rev.
Française Automat. Informat. Recherche Opérationnelle
Sér. Rouge 8 (1974), no. R2, 129–151
(English, with loose French summary). MR 0365287
(51 #1540)
 [6]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., Amsterdam, 1978. Studies in Mathematics and
its Applications, Vol. 4. MR 0520174
(58 #25001)
 [7]
G.
Duvaut and J.L.
Lions, Inequalities in mechanics and physics, SpringerVerlag,
Berlin, 1976. Translated from the French by C. W. John; Grundlehren der
Mathematischen Wissenschaften, 219. MR 0521262
(58 #25191)
 [8]
I. Hlavaček, "Convergence of an equilibrium finite element method for plane elastostatics," Apt. Math., v. 24, 1979, pp. 427456.
 [9]
C.
Johnson and B.
Mercier, Some equilibrium finite element methods for
twodimensional elasticity problems, Numer. Math. 30
(1978), no. 1, 103–116. MR 0483904
(58 #3856)
 [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
(83d:65287), http://dx.doi.org/10.1090/S00255718198206456572
 [11]
D. G. Luenberger, Introduction to Linear and Nonlinear Programming, AddisonWesley, Reading, Mass., 1973.
 [12]
D. Malkus & T. Hughes, "Mixed finite element methodsreduced and selective integration techniques: A unification of concepts," Comp. Methods Appl. Mech. Engrg., v. 15, 1978, pp. 6381.
 [13]
J.
A. Nitsche, On Korn’s second inequality, RAIRO Anal.
Numér. 15 (1981), no. 3, 237–248
(English, with French summary). MR 631678
(83a:35012)
 [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. 167207.
 [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
(80f:73063)
 [16]
V. B. Watwood, Jr. & B. J. Hartz, "An equilibrium stress field model for finite element solutions of twodimensional elastostatic problems," Internat. J. Solids and Structures, v. 4, 1968, pp. 857873.
 [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. 145197.
 [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. 224258.
 [1]
 I. Babuška, "Error bounds for finite element method," Numer. Math., v. 16, 1971, pp. 322333. MR 0288971 (44:6166)
 [2]
 I. Babuška, J Osborn & J. Pitkäranta, "Analysis of mixed methods using mesh dependent norms," Math. Comp., v. 35, 1980, pp. 10391062. MR 583486 (81m:65166)
 [3]
 M. Bercovier, "Perturbation of mixed variational problems. Application to mixed finite element methods," RAIRO Anal. Numér., v. 12, 1978, pp. 211236. MR 509973 (80b:49031)
 [4]
 J. Bramble & S. Hilbert, "Estimation of linear functional on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112124. MR 0263214 (41:7819)
 [5]
 F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers," RAIRO Ser. Rouge, v. 8, 1974, pp. 129151. MR 0365287 (51:1540)
 [6]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [7]
 G. Duvaut & J. L. Lions, Inequalities in Mechanics and Physics, SpringerVerlag, Berlin and New York, 1976. MR 0521262 (58:25191)
 [8]
 I. Hlavaček, "Convergence of an equilibrium finite element method for plane elastostatics," Apt. Math., v. 24, 1979, pp. 427456.
 [9]
 C. Johnson & B. Mercier, "Some equilibrium finite element methods for twodimensional elasticity problems," Numer. Math., v. 30, 1978, pp. 103116. MR 0483904 (58:3856)
 [10]
 C. Johnson & J. Pitkäranta, "Analysis of some mixed finite element methods related to reduced integration," Math. Comp, v. 38, 1982, pp. 375400. MR 645657 (83d:65287)
 [11]
 D. G. Luenberger, Introduction to Linear and Nonlinear Programming, AddisonWesley, Reading, Mass., 1973.
 [12]
 D. Malkus & T. Hughes, "Mixed finite element methodsreduced and selective integration techniques: A unification of concepts," Comp. Methods Appl. Mech. Engrg., v. 15, 1978, pp. 6381.
 [13]
 J. Nitsche, "On Korn's second inequality," RAIRO Anal. Numér., v. 15, 1981, pp. 237248. MR 631678 (83a:35012)
 [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. 167207.
 [15]
 R. Taylor & O. C. Zienkiewicz, "Complementary energy with penalty functions in finite element analysis," in Energy Methods in Finite Element Analysis (R. Glowinski, E. Y. Rodin and O. C. Zienkiewicz, eds.), Wiley, New York, 1979, pp. 143174. MR 537004 (80f:73063)
 [16]
 V. B. Watwood, Jr. & B. J. Hartz, "An equilibrium stress field model for finite element solutions of twodimensional elastostatic problems," Internat. J. Solids and Structures, v. 4, 1968, pp. 857873.
 [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. 145197.
 [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. 224258.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819830717693X
PII:
S 00255718(1983)0717693X
Article copyright:
© Copyright 1983 American Mathematical Society
