Superconvergence and reduced integration in the finite element method
Author:
Miloš Zlámal
Journal:
Math. Comp. 32 (1978), 663685
MSC:
Primary 65N30; Secondary 65D30
MathSciNet review:
0495027
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Abstract: The finite elements considered in this paper are those of the Serendipity family of curved isoparametric elements. There is given a detailed analysis of a superconvergence phenomenon for the gradient of approximate solutions to second order elliptic boundary value problems. An approach is proposed how to use the superconvergence in practical computations.
 [1]
J. BARLOW, "Optimal stress locations in finite element models," Internat. J. Numer. Methods., v. 10, 1976, pp. 243251.
 [2]
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)
 [3]
J.
H. Bramble and S.
R. Hilbert, Bounds for a class of linear functionals with
applications to Hermite interpolation, Numer. Math.
16 (1970/1971), 362–369. MR 0290524
(44 #7704)
 [4]
P.
G. Ciarlet and P.A.
Raviart, The combined effect of curved boundaries and numerical
integration in isoparametric finite element methods, The mathematical
foundations of the finite element method with applications to partial
differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md.,
1972), Academic Press, New York, 1972, pp. 409–474. MR 0421108
(54 #9113)
 [5]
Bruce
M. Irons and Abdur
Razzaque, Experience with the patch test for convergence of finite
elements, The mathematical foundations of the finite element method
with applications to partial differential equations (Proc. Sympos., Univ.
Maryland, Baltimore, Md., 1972), Academic Press, New York, 1972,
pp. 557–587. MR 0423839
(54 #11813)
 [6]
J. NEČAS, Les Méthodes Directes en Théorie des Equations Elliptiques, Academia, Prague, 1967.
 [7]
D. A. VERYARD, Problems Associated with the Convergence of Isoparametric and Mixoparametric Finite Elements, M. Sc. Thesis, University of Wales, 1971.
 [8]
O.
C. Zienkiewicz, The finite element method in engineering
science, McGrawHill, London, 1971. The second, expanded and revised,
edition of The finite element method in structural and continuum mechanics.
MR
0315970 (47 #4518)
 [9]
Miloš
Zlámal, Some superconvergence results in the finite element
method, Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin,
1977, pp. 353–362. Lecture Notes in Math., Vol. 606. MR 0488863
(58 #8365)
 [1]
 J. BARLOW, "Optimal stress locations in finite element models," Internat. J. Numer. Methods., v. 10, 1976, pp. 243251.
 [2]
 J. H. BRAMBLE & S. R. HILBERT, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 113124. MR 0263214 (41:7819)
 [3]
 J. H. BRAMBLE & S. R. HILBERT, "Bounds for a class of linear functionals with applications to Hermite interpolation," Numer. Math., v. 16, 1971, pp. 362369. MR 0290524 (44:7704)
 [4]
 P. G. CIARLET & P. A. RAVIART, "The combined effect of curved boundaries and numerical integration in isoparametric finite element methods," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 409474. MR 0421108 (54:9113)
 [5]
 B. M. IRONS & A. RAZZAQUE; "Experience with the patch test for convergence of finite elements," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 557587. MR 0423839 (54:11813)
 [6]
 J. NEČAS, Les Méthodes Directes en Théorie des Equations Elliptiques, Academia, Prague, 1967.
 [7]
 D. A. VERYARD, Problems Associated with the Convergence of Isoparametric and Mixoparametric Finite Elements, M. Sc. Thesis, University of Wales, 1971.
 [8]
 O. C. ZIENKIEWICZ, The Finite Element Method in Engineering Science, McGrawHill, London, 1972. MR 0315970 (47:4518)
 [9]
 M. ZLÁMAL, "Some superconvergence results in the finite element method," Mathematical Aspects of Finite Element Methods, SpringerVerlag, Berlin, Heidelberg, New York, 1977, pp. 351362. MR 0488863 (58:8365)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804950274
PII:
S 00255718(1978)04950274
Keywords:
Finite elements
Article copyright:
© Copyright 1978 American Mathematical Society
