Superconvergence and reduced integration in the finite element method

Author:
Miloš Zlámal

Journal:
Math. Comp. **32** (1978), 663-685

MSC:
Primary 65N30; Secondary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1978-0495027-4

MathSciNet review:
0495027

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Abstract | References | Similar Articles | Additional Information

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. 243-251.**[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**, https://doi.org/10.1137/0707006**[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**, https://doi.org/10.1007/BF02165007**[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****[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****[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*, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR**0315970****[9]**Miloš Zlámal,*Some superconvergence results in the finite element method*, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 353–362. Lecture Notes in Math., Vol. 606. MR**0488863**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0495027-4

Keywords:
Finite elements

Article copyright:
© Copyright 1978
American Mathematical Society