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A variational principle for adaptive approximation of ordinary differential equations

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Summary

A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error=∑local error⋅weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms.

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References

  1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142, 1–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alekseev, V.M.: An estimate for the perturbations of the solutions of ordinary differential equations. II. Vestnik Moskov. Univ. Ser. I Mat. Mech 3, 3–10 (1961)

    Google Scholar 

  3. Babuška, I., Miller, A., Vogelius, M.: Adaptive methods and error estimation for elliptic problems of structural mechanics. In: Adaptive computational methods for partial differential equations. (SIAM, Philadelphia, Pa., 1983) 57–73

  4. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4(4), 237–264 (1996)

    MATH  Google Scholar 

  5. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 1–102 (2001)

  6. Böttcher, K., Rannacher, R.: Adaptive error control in solving ordinary differential equations by the discontinuous Galerkin method. Preprint, 1996

  7. Dahlquist, G., Björk, Å197.: Numerical Methods. (Prentice-Hall, 1974)

  8. Dahlquist, G., Björk, Å197.: Numerical Mathematics. http://www.mai.liu.se/∼akbjo/NMbook.html.

  9. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numerica 105–158 (1995)

  10. Estep, D.: A posteriori error bounds and global error control for approximation of ordinary differential equations. SIAM J. Numer. Anal. 32, 1–48 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Gröbner, W.: Die Lie-Reihen und ihre Anwendungen, (VEB Deutscher Verlag der Wissenschaften, Berlin, 1967)

  12. Harrier, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. (Springer-Verlag, 1993)

  13. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. (John Wiley & Sons, Inc., 1962)

  14. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. (Society for Industrial and Applied Mathematics, 1996)

  15. Johnson, C.: Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 25, 908–926 (1988)

    MathSciNet  MATH  Google Scholar 

  16. Johnson, C., Szepessy, A.: Adaptive finite element methods for conservation laws based on a posteriori error estimates. Comm. Pure Appl. Math. 48, 199–234 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Moon, K.-S., Szepessy, A., Tempone, R., Zouraris, G.E.: Hyperbolic differential equations and adaptive numerics. In: Theory and numerics of differential equations (Eds. J.F. Blowey, J.P. Coleman and A.W. Craig, Durham 2000, Springer Verlag, 2001)

  18. Moon, K.-S., Szepessy, A., Tempone, R., Zouraris, G.E. (2003) Convergence rates for adaptive approximation of ordinary differential equations. Numer. Math. (in press). DOI 10.1007/s00211-003-0466-9

  19. Prothero, A.: Estimating the accuracy of numerical solutions to ordinary differential equations. In: Computational Techniques for Ordinary Differential Equations. I. Gladwell and D.K. Sayers, eds. Academic Press, New York, 1980, pp. 103–128

  20. Utumi, T., Takaki, R., Kawai, T.: Optimal time step control for the numerical solution of ordinary differential equations. SIAM J. Numer. Anal. 33, 1644–1653 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institutionen för Numerisk Analys och Datalogi, Kungl Tekniska Högskolan, 100 44, Stockholm, Sweden

    Kyoung-Sook Moon, Anders Szepessy & Raúl Tempone

  2. Matematiska Institutionen, Kungl Tekniska Högskolan, 100 44, Stockholm, Sweden

    Anders Szepessy

  3. Department of Mathematics, IACM, FO.R.T.H, University of the Aegean, 832 00, 711 10, Karlovassi, Heraklion, Samos, Greece

    Georgios E. Zouraris

Authors
  1. Kyoung-Sook Moon
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  2. Anders Szepessy
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  3. Raúl Tempone
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  4. Georgios E. Zouraris
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Correspondence to Kyoung-Sook Moon.

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Mathematics Subject Classification (2000): 65L70, 65G50

This work has been supported by the EU–TMR project HCL # ERBFMRXCT960033, the EU–TMR grant # ERBFMRX-CT98-0234 (Viscosity Solutions and their Applications), the Swedish Science Foundation, UdelaR and UdeM in Uruguay, the Swedish Network for Applied Mathematics, the Parallel and Scientific Computing Institute (PSCI) and the Swedish National Board for Industrial and Technical Development (NUTEK).

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Moon, KS., Szepessy, A., Tempone, R. et al. A variational principle for adaptive approximation of ordinary differential equations . Numer. Math. 96, 131–152 (2003). https://doi.org/10.1007/s00211-003-0467-8

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  • DOI: https://doi.org/10.1007/s00211-003-0467-8

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