Abstract
The approximation involved in discretizing continuum problems by the finite element method may result in unacceptable errors. Much effort has been directed to keeping such errors under control so that engineering decision can be confidently made. Indeed, ideally the user will seek a solution with an a-priori specific accuracy tolerance to avoid excessive cost of providing an over-accurate solution.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. Babuska and W. C. Rheinboldt, Error Estimates for Adaptive Finite Element Computations, SIAM J. Numer. Anal. Vol.15 (1978), pp.736–754.
I. Babuska and W. C. Rheinboldt, A-posteriori Error Estimates for the Finite Element Method, Int. J. Num. Meth. Engng., Vol.12 (1978) pp. 1597–1615.
D. W. Kelly, J. P. de S. R. Gago, O. C. Zienkiewicz and I. Babuska, A-posteriori Error Analysis and Adaptive Processes in the Finite Element Method: Part I — Error Analysis, Int. J. Num. Meth. Engng. Vol.19 (1983), pp.1593–1619.
D. W. Kelly, The Self-equilibration of Residuals and Complementary A-posteriori Error Estimates in the Finite Element Method, Int. J. Num. Meth. Engng. Vol.20 (1984), pp. 1491–1506.
D. W. Kelly, R. J. Mills, J. A. Reizes and A. D. Miller, A-posteriori Estimates of the Solution Error caused by Discretization in the Finite Element, Finite Difference and Boundary Element Methods, Int. J. Num. Meth. Engng. Vol.24 (1987), pp. 1921–1939.
M. Ainsworth and A. Craig, A-posteriori Error Estimators in the Finite Element Methods, Numer. Math. Vol.60 (1992), pp.429–463.
M. Ainsworth and J. T. Oden, A Unified Approach to A-posteriori Error Estimation using Element Residual Methods, Numer. Math. (in press).
O. C. Zienkiewicz, J. P. de S. R. Gago and D. W. Kelly, The Hierarchical Concept in Finite Element Analysis, Comp. Struct. Vol.16, (19), pp.53–65.
O. C. Zienkiewicz and J. Z. Zhu, A Simple Error Estimator and the Adaptive Procedure for Practical Engineering Analysis, Int. J. Num. Meth. Engng. Vol. 24 (1987), pp.337–357.
I. Babuska and R. Rodriguez, The Problem of the Selection of an A-posteriori Error Indicator based on Smoothening Techniques, Int. J. Num. Meth. Engng. Vol. 36 (1993), pp.539–567.
O. C. Zienkiewicz and J. Z. Zhu, The Superconvergent Patch Recovery and A-posteriori Error Estimates. Part 1: The Recovery Technique, Int. J. Num. Meth. Engng. Vol.33 (1992), pp. 1331–1364.
O. C. Zienkiewicz and J. Z. Zhu, The Superconvergent Patch Recovery and A-posteriori Error Estimates. Part 2: Error Estimates and Adaptivity, Int. J. Num. Meth. Engng. Vol.33 (1992), pp. 1365–1382.
O. C. Zienkiewicz and J. Z. Zhu, The Superconvergent Patch Recovery (SPR) and Adaptive Finite Element Refinement, Comput. Meth. Appl. Mech. Engng, Vol.101 (1992), pp.207–224.
O. C. Zienkiewicz, J. Z. Zhu and J. Wu, Superconvergent Patch Recovery Techniques. Some Further Tests, Comm. Num. Meth. Engng. Vol.9 (1993), pp.251–258.
I. Babuska, T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj and K. Copps, An Objective Criterion for Assessing the Reliability of A-posteriori Error Estimators in Finite Element Computations, USACM Bulletin, Vol. 7 (1994), pp.4–15.
I. Babuska, T. Strouboulis and C. S. Upadhyay, A Model Study of the Quality of A-posteriori Error Estimators for Linear Elliptic Problems. Error Estimation in the interior of Patchwise Uniform Grids of Triangles, Comput. Meth. Appl. Mech. Engng. Vol. 114 (1994), pp.307–378.
I. Babuska, T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj and K. Copps, Validation of A-posteriori Error Estimators by Numerical Approach, Int. J. Num. Meth. Engng. Vol.37 (1994), pp. 1073–1123.
N-E. Wiberg, Enhanced Superconvergent Patch Recovery incorporating Equilibrium and Boundary Conditions, Int. J. Num. Meth. Engng. Vol.37 (1994), pp. 3417–3449.
N-E. Wiberg and F. Adulwahab, An Efficient Post-processing Technique for Stress Problems based on Superconvergent Derivatives and Equilibrium, Numerical Methods in Engineering (Ch. Hirsch et al. eds), pp. 25–32, Elsevier, Amsterdam. (1992)
T. Blacker and T. Belytschko, Superconvergent Patch Recovery with Equilibrium and Conjoint Interpolant Enhancements, Int. J. Num. Meth. Engng. Vol.37 (1994), pp.517–536.
M. Tabbara, T. Blacker and T. Belytschko, Finite Element Derivative Recovery by moving Least Square Interpolants, Comput. Meth. Appl. Mech. Engng. Vol.117 (1994), pp. 211–223.
J. Peraire, M. Vahdati, K. Morgan and O. C. Zienkiewicz, Adaptive Remeshing for Compressible Flow Computations, J. Comp. Phys. Vol.72 (1987), pp. 449–466.
J. Peraire, J. Peiro, L. Fomaggia, K. Morgan and O. C. Zienkiewicz, Finite Element Euler Computations in Three Dimensions, Int. J. Num. Meth. Engng. Vol.26 (1988), pp. 2135–2159.
J. Z. Zhu, O. C. Zienkiewicz, E. Hinton and J. Wu, A New Approach to the Development of Automatic Quadrilateral Mesh Generation, Int. J. Num. Meth. Engng. Vol.32, (1991), pp. 849–866.
N. P. Weatherill and M. S. Sheperd (eds) Special issue on Adaptive Meshing, Int. J. Num. Meth. Engng. Vol.32, (1991), pp. 651–939.
O. C. Zienkiewicz, Y. C. Liu and G. C. Huang, Error Estimation and Adaptivity in Flow Formulation for Forming Problems, Int. J. Num. Meth. Engng. Vol.25 (1988), pp. 23–42.
O. C. Zienkiewicz, G. C. Huang and Y. C. Liu, Adaptive FEM Computation of Forming Processes — Application to Porous and Nonporous Materials, Int. J. Num. Meth. Engng., Vol. 28 (1989), pp. 1327–1554.
O. C. Zienkiewicz and G. C. Huang, A Note on Localisation Phenomena and Adaptive Finite Element Analysis in Forming Processes, Comm. Appl. Numer. Mesh, Vol.6 (1990), pp. 71–76.
J. Wu, J. Z. Zhu, J. Szmelter and O. C. Zienkiewicz, Error Estimation and Adaptivity in Navier Stokes Incompressible Flows, Comp. Mesh. Vol.6 (1990), pp. 259–270.
O. C. Zienkiewicz and Y. M. Xie, A Simple Error Estimator and Adaptive Time Stepping Procedure for Dynamic Analysis, Earthquake Eng. Struct. Dynamics, Vol.22 (1991), pp. 871–887.
E. Oñate and J. Castro, Adaptive Mesh Refinement Techniques for Structural Problems, in “The Finite Element Method in the 90′s”, E. Oñate, J. Periaux and A. Samuelsson (Eds.), Springer-Verlag/CIMNE (1991).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zienkiewicz, O.C., Zhu, J.Z. (1995). Error estimation and Adaptivity: Achievements of the last decade. In: Atluri, S.N., Yagawa, G., Cruse, T. (eds) Computational Mechanics ’95. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79654-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-79654-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-79656-2
Online ISBN: 978-3-642-79654-8
eBook Packages: Springer Book Archive