×

zbMATH — the first resource for mathematics

The superconvergent patch recovery and \(a\;posteriori\) error estimates. II: Error estimates and adaptivity. (English) Zbl 0769.73085
Summary: The issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz-Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper [see the foregoing entry] is used in the computation of the Zienkiewicz-Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.

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. 33 pp 1331– (1992)
[2] Zienkiewicz, Int. j. numer. methods eng. 24 pp 337– (1987)
[3] Babuska, SIAM J. Numer. Anal. 15 pp 736– (1978)
[4] Kelly, Int. j. numer. methods eng. 19 pp 1593– (1983)
[5] Gago, Int. j. numer. methods eng. 19 pp 1621– (1983)
[6] Zienkiewicz, Comp. Struct. 16 pp 53– (1983)
[7] Bank, Math. Comp. 44 pp 283– (1985)
[8] Oden, Comp. Methods Appl. Mech. Eng. 77 pp 113– (1989)
[9] Zienkiewicz, Int. j. numer. methods eng. 28 pp 2839– (1989)
[10] Zienkiewicz, Int. j. numer. methods eng. 25 pp 23– (1988)
[11] Zienkiewicz, Int. j. numer. methods eng. 28 pp 2191– (1989)
[12] Wu, Comp. Mech. 6 pp 259– (1990)
[13] Zhu, Int. j. numer. methods eng. 30 pp 1321– (1990)
[14] Zienkiewicz, Comp. Methods Appl. Mech. Eng. 82 pp 95– (1990)
[15] Zienkiewicz, Int. j. numer. methods eng. 32 pp 783– (1991)
[16] Zienkiewicz, Int. j. numer. methods eng. 28 pp 879– (1989)
[17] Hinton, Int. j. numer. methods eng. 8 pp 461– (1974)
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.