Can We Trust the Computational Analysis of Engineering Problems?

  • I. Babuška
  • T. Strouboulis
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 19)

Abstract

Computational science in general and computational mechanics in particular addresses physical and engineering reality with respect to some particular goals. These goals must be clearly specified. They are usually to get good qualitative or quantitative information about reality. The admissible quality of required information should be characterized.

Keywords

Finite Element Method Posteriori Error Effectivity Index Posteriori Error Estimate Finite Element Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Roache, P.J.: Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, 1998.Google Scholar
  2. [2]
    Babuška, L, Chleboun, J.: Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions, Preprint, TICAM University of Texas at Austin, 2000.Google Scholar
  3. [3]
    Babuška, I., Strouboulis T.: Finite Element Method and its Reliablity, Oxford University Press, 2001.Google Scholar
  4. [4]
    Babuška, I., Rheinboldt, W.C.: A-posteriori bounds and adaptive procedures for the finite element method, Recent advances in engineering science, Proceedings of 15the annual meeting, Soc. of Engg. Science, University of Florida, Decision of Continuing Education, 1978, 413–423.Google Scholar
  5. [5]
    Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 1978, 736–754.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Babuška, I., Miller, A.: A-posteriori error estimates and adaptive techniques for the finite element method, Technical Note BN-968, Institute for Physical Science and Technology, University of Maryland, 1981.Google Scholar
  7. [7]
    Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation: Part 1. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg., 61, 1987, 1–40.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Verfürth, R.: A-posteriori error estimators for singularly perturbed reaction diffusion equations, Num. Math., 78, 1998, 479–493.MATHCrossRefGoogle Scholar
  9. [9]
    Cartensen, C., Funken, S.A.: Fully reliable localized error control in FEM, Ber. de. Math., Seminar Kiel, 97–12, 1992.Google Scholar
  10. [10]
    Morin, P., Nochetto, R.H., Siebert, G.: Preprint, 2000.Google Scholar
  11. [11]
    Babuška, I., Rheinboldt, W. C.: A-posteriori error estimates for the finite element method, Int. J. Numer. Methods Engrg., 12, 1978, 1597–1615.MATHCrossRefGoogle Scholar
  12. [12]
    Babuška, L, Duran, R., Rodriguez, R.: Analysis of the efficiency of an a-posteriori error estimator for linear triangular elements, SIAM J. Numer. Anal., 29, 1992, 947–964.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Cartensen, C., Funken, S.A.: Constants in Clement interpolation error and residual based a-posteriori estimates in finite element method, Ber. de. Math., Seminar Kiel, 97–11, 1997.Google Scholar
  14. [14]
    Ladeveze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20, 1983, 485–509.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 1985, 283–301.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Ainsworth, M., Oden, J.T.: A-posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg.: Computaional Mechanics Advances, 142, 1997, 1–88.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Ainsworth, M., Oden, J. T.: A unified approach to a-posteriori error estimation using element residual methods, Numer. Math., 65, 1993, 23–50.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Oden, J. T., Demkowicz, L., Rachowicz, W., Westermann, T. A.: Toward a universal h-p adaptive finite element strategy: Part 2, A Posteriori Error Estimates, Comput. Methods Appl. Mech. Engrg., 77, 1989, 113–180.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Zienkiewicz, O. C., Zhu, J. Z.: A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Engrg., 24, 1987, 337–357.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Zienkiewicz, O. C., Zhu, J. Z.: The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Engrg., 33, 1992, 1331–1364.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Zienkiewicz, O. C., Zhu, J. Z.: The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Engrg., 33, 1992, 1365–1382.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Ainsworth, M., Oden, J.T.: A-posteriori error estimation in Finite Element Analysis, John Wiley, 2000.Google Scholar
  23. [23]
    Verfürth, R.: A review of a-posteriori error estimation and adaptive meshrefinement techniques, Wiley, Teubner, 1996.Google Scholar
  24. [24]
    Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations, Acta. Numer., Cambridge Univ. Press, 1995, 105–158.Google Scholar
  25. [25]
    Babuška, L, Strouboulis, T., Upadhyay, C. S.: 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. Methods Appl. Mech. Engrg., 114, 1994, 307–378.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Babuška, I., Strouboulis, T., Upadhyay, C. S.: A model study of the quality of a-posteriori error estimators for finite element solutions of linear elliptic problems with particular reference to the behaviour near the boundary, Int. J. Numer. Methods Engrg., 40, 1997, 2521–2577.MATHCrossRefGoogle Scholar
  27. [27]
    Babuška, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, S. K., Copps, K.: An objective criterion for assessing the reliability of a-posteriori error estimators in finite element computation, IACM bulletin, 9, 1994, 27–37.Google Scholar
  28. [28]
    Babuška, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, S. K.: Computer based proof of the existence of superconvergence points in the finite element method: Superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson and elasticity equation, Num. Meth. for PDE’s, 12, 1996, 347–392.MATHCrossRefGoogle Scholar
  29. [29]
    Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, Springer-Verlag, 1995.Google Scholar
  30. [30]
    Babuška, I., Miller, A.: The post-processing approach in the finite element method-Part 3: A-posteriori error estimates and adaptive mesh selection, Int. J. Numer. Methods Engrg., 20, 1984, 2311–2324.MATHCrossRefGoogle Scholar
  31. [31]
    Babuška, I., Strouboulis, T., Gangaraj, S. K.: Guaranteed computable bounds for the exact error in the finite element solution-Part 1: One dimensional model problem, Comput. Methods Appl. Mech. Engrg., 176, 1999, 51–79.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    Prudhomme, S., Oden, J. T.: On goal oriented error estimation for elliptic problems: Applications to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg., 176, 1999, 313–331.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Paraschivoiu, M., Peraire, J., Patera, A. T.: A-posteriori finite element bounds for linear functional outputs of elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg., 150, 1997, 289–312.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Becker, R., Rannacher, R.: A feedback approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4, 1996, 237–264.MathSciNetMATHGoogle Scholar
  35. [35]
    Deb, M.K., Babuška, I., Oden, J.T.: Solution of Stochastic Partial Differential equations using Galerkin method and finite element techniques, Preprint, 2000.Google Scholar
  36. [36]
    Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach, Springer, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • I. Babuška
    • 1
  • T. Strouboulis
    • 2
  1. 1.Texas Institute for Computational and Applied MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations