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Convergence rates for adaptive approximation of ordinary differential equations

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This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by \(\mathcal O(N)\). The alternative version with constant tolerance stops with \(\mathcal O(N\ {\rm log}\ N)\) total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the \({{L^{{\frac{{1}}{{p+1}}}}}}\) quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L 1-norm of the error density.

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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. 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, pp. 57–73

  3. Babuška, I., Vogelius, M.: Feedback and adaptive finite element solution of one- dimensional boundary value problems. Numer. Math. 44(1), 75–102 (1984)

    Google Scholar 

  4. Bakhvalov, N.S.: On the optimality of linear methods for operator approximation in convex classes of functions. USSR Comput. Math. and Math. Phys. 11, 244–249 (1971)

    Google Scholar 

  5. 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 

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

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

  8. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp. 70(233), 25–75 (2001)

    Google Scholar 

  9. Dahlquist, G., Björk, Å.: Numerical Methods. Prentice-Hall, 1974

  10. DeVore, R.A.: Nonlinear approximation, Acta Numerica, 51–150 (1998)

  11. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

    Article  MATH  Google Scholar 

  12. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Google Scholar 

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

  14. 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 

  15. Estep, D., Hodges, D., Warner, M.: Computational error estimates and adaptive error control for a finite element solution of launch vehicle trajectrory problems. SIAM J. Sci. Comput. 21, 1609–1631 (2000)

    MATH  Google Scholar 

  16. Estep, D., Johnson, C.: The pointwise computability of the Lorenz system. Math. Models Methods Appl. Sci. 8, 1277–1305 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Gao, F.: Probabilistic analysis of numerical integration algorithms. J. Complexity 7(1), 58–69 (1991)

    MATH  Google Scholar 

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

  19. Higham, D.J., Higham, N.J.: Matlab Guide, (Society for Industrial and Applied Mathematics, 2000)

  20. 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 

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

    MathSciNet  MATH  Google Scholar 

  22. Lamba, H., Stuart, A.M.: Convergence results for the MATLAB ODE23 routine. BIT 38(4), 751–780 (1998)

    MATH  Google Scholar 

  23. Logg, A.: Multi-adaptive Galerkin methods for ODEs. (Licentiate thesis, ISSN 0347–2809, Chalmers University of Technology, 2001). http://www.md.chalmers.se/Centres/Phi/preprints/index.html.

  24. Lorenz, E.N.: Deterministic non-periodic flows. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  25. Moon, K.-S.: Convergence rates of adaptive algorithms for deterministic and stochastic differential equations. (Licentiate thesis, ISBN 91-7283-196-0, Royal Institute of Technology, 2001). http://www.nada.kth.se/∼moon/paper.html.

  26. Moon, K.-S., von Schwerin, E., Szepessy, A., Tempone, R.: Convergence rates for adaptive finite element approximation of partial differential equations work in progress.

  27. 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, A.W. Craig, Durham 2000, Springer Verlag, 2001)

  28. Moon, K.-S., Szepessy, A., Tempone, R., Zouraris, G.E. (2003) A variational principle for adaptive approximation of ordinary differential equations. Numer. Math. (in press). DOI 10.1007/s00211-003-0467-8

  29. Morin, P., Nochetto, R., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Novak, E.: On the power of adaption. J. Complexity 12, 199–237 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Stuart, A.M.: Probabilistic and detereministic convergence proofs for software for initial value problems. Numerical Algorithms 14, 227–260 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Szepessy, A., Tempone, R.: Optimal control with multigrid and adaptivity. Fifth World Congress on Computational Mechanics http://wccm.tuwien.ac.at.

  33. Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54, 1169–1214 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Söderlind, G.: Automatic control and adaptive time-stepping. ANODE01 Proceedings, Numerical Algorithms

  35. Traub, J.F., Werschulz, A.G.: Complexity and Information. Cambridge University Press, Cambridge, 1998

  36. 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 

  37. Werschulz, A.G.: The Computational Complexity of Differential and Integral Equations, An Information-Based Approach. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991

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

    Georgios E. Zouraris

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  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): 65Y20, 65L50, 65L70

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. Convergence rates for adaptive approximation of ordinary differential equations. Numer. Math. 96, 99–129 (2003). https://doi.org/10.1007/s00211-003-0466-9

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  • DOI: https://doi.org/10.1007/s00211-003-0466-9

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