×

zbMATH — the first resource for mathematics

The superconvergent patch recovery and \(a\;posteriori\) error estimates. I: The recovery technique. (English) Zbl 0769.73084
A general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an \(O(h^ 4)\) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zienkiewicz, Int. j. numer. methods eng. 24 pp 337– (1987)
[2] Ainsworth, Int. j. numer. methods eng. 28 pp 2161– (1989)
[3] Zhu, Int. j. numer. methods eng. 30 pp 1321– (1990)
[4] Zienkiewicz, Int. j. numer. methods eng. 32 pp 783– (1991)
[5] and , The Finite Element Method, 4th edn, Vol. I McGraw-Hill, New York, 1989.
[6] Moan, ZAMM 54 pp 501– (1974)
[7] Barlow, Int. j. numer. methods eng. 10 pp 243– (1976)
[8] Zlamal, Math. Comp. 32 pp 663– (1978)
[9] Lesaint, RAIRO Anal. Numer. 13 pp 139– (1979)
[10] Chen, Numer. Math. J. Chinese Univ. 3 pp 118– (1981)
[11] Mackinnon, Int. j. numer. methods eng. 28 pp 489– (1989)
[12] Chen, Numer. Math. J. Chinese Univ. 2 pp 12– (1980)
[13] Levine, Numer. Anal. 5 pp 407– (1985)
[14] Zhu, Numer. Math. J. Chinese Univ. 5 pp 311– (1983)
[15] Andreev, Numer. Methods P.D.E.’s 4 pp 15– (1986)
[16] Wheeler, Numer. Methods Partial Differential Equations 3 pp 65– (1987)
[17] Hinton, Int. j. numer. methods eng. 8 pp 461– (1974)
[18] Krizek, Acta Appl. Math. 9 pp 175– (1987)
[19] Oden, Int. j. numer. methods eng. 3 pp 317– (1971)
[20] Oden, Int. j. numer. methods eng. 6 pp 55– (1973)
[21] Zienkiewicz, Commun. appl. numer. methods 1 pp 3– (1985)
[22] Babuska, Int. j. numer. methods eng. 20 pp 1085– (1984)
[23] Bramble, Math. Comp. 31 pp 94– (1977)
[24] Thomee, Math. Comp. 31 pp 652– (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.