Skip to main content
SpringerLink
Log in

An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We investigate an h-p version of the continuous Petrov–Galerkin time stepping method for nonlinear delay differential equations with vanishing delays. We derive a priori error estimates in the \(L^{2}\)-, \(H^{1}\)- and \(L^\infty \)-norm that are completely explicit with respect to the local time steps, the local polynomial degrees, and the local regularity of the exact solution. Moreover, we show that the h-p version continuous Petrov–Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. The theoretical results are illustrated by some numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications Inc., New York (1992)

  2. Adams, R.A.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

    Google Scholar 

  3. Ali, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27, 254–265 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Suri, M.: The \(p\) and \(h\)-\(p\) versions of the finite element method, basic principles and properties. SIAM Rev. 36, 578–632 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker, C.T.H., Paul, C.A.H., Willé, D.R.: A bibliography on the numerical solution of delay differential equations, NA Report 269, Dept of Mathematics, University of Manchester (1995)

  6. Bellen, A.: One-step collocation for delay differential equations. J. Comput. Appl. Math. 10, 275–283 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellen, A., Zennaro, M.: Numerical solution of delay differential equations by uniform corrections to an implicit Runge–Kutta method. Numer. Math. 47, 301–316 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  9. Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  10. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  11. Brunner, H., Huang, Q., Xie, H.: Discontinuous Galerkin methods for delay differential equations of pantograph type. SIAM J. Numer. Anal. 48, 1944–1967 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brunner, H., Schötzau, D.: \(hp\)-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J. Numer. Anal. 44, 224–245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng, K., Xiong, Z., Huang, Y.: The Galerkin continuous finite element method for delay-differential equation with a variable term. Appl. Math. Comput. 186, 1488–1496 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Engelborghs, K., Luzyanina, T., In’t Hout, K.J., Roose, D.: Collocation methods for the computation of periodic solutions of delay differential equations. SIAM J. Sci. Comput. 22, 1593–1609 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Griffiths, D.F., Lorenz, J.: An analysis of the Petrov–Galerkin finite element method. Comput. Methods Appl. Mech. Eng. 14, 39–64 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Q.M., Xie, H.H., Brunner, H.: The \(hp\) discontinuous Galerkin method for delay differential equations with nonlinear vanishing delay. SIAM J. Sci. Comput. 35, A1604–A1620 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, Q.M., Xu, X.X., Brunner, H.: Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete Contin. Dyn. Syst. 36, 5423–5443 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ilyin, A., Laptev, A., Loss, M., Zelik, S.: One-dimensional interpolation inequalities, Carlson-Landau inequalities, and magnetic Schrödinger operators. Int. Math. Res. Not. 2016, 1190–1222 (2016)

    Article  MATH  Google Scholar 

  19. Ito, K., Tran, H.T., Manitius, A.: A fully-discrete spectral method for delay-differential equations. SIAM J. Numer. Anal. 28, 1121–1140 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45, 285–312 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, D., Zhang, C.: Superconvergence of a discontinuous Galerkin Method for first-order linear delay differential equations. J. Comput. Math. 29, 574–588 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maset, S.: Stability of Runge–Kutta methods for linear delay differential equations. Numer. Math. 87, 355–371 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schötzau, D., Schwab, C.: An \(hp\) a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37, 207–232 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schwab, C.: p- and hp- Finite Element Methods. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  25. Shen, J., Wang, L.L.: Legendre and Chebyshev dual-Petrov–Galerkin methods for hyperbolic equations. Comput. Methods Appl. Mech. Eng. 196, 3785–3797 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Takama, N., Muroya, Y., Ishiwata, E.: On the attainable order of collocation methods for delay differential equations with proportional delay. BIT Numer. Math. 40, 374–394 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tang, J., Xie, Z.Q., Zhang, Z.M.: The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete Contin. Dyn. Syst. Ser. B 18, 797–819 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Tian, H.J., Yu, Q.H., Kuang, J.X.: Asymptotic stability of linear neutral delay differential-algebraic equations and Runge–Kutta methods. SIAM J. Numer. Anal. 52, 68–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, Z.Q., Wang, L.L.: A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete Contin. Dyn. Syst. Ser. B 13, 685–708 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wei, Y.X., Chen, Y.P.: Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. J. Sci. Comput. 53, 672–688 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wihler, T.P.: An a priori error analysis of the \(hp\)-version of the continuous Galerkin FEM for nonlinear initial value problems. J. Sci. Comput. 25, 523–549 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wihler, T.P.: A note on a norm-preserving continuous Galerkin time stepping scheme. Calcolo (2016). doi:10.1007/s10092-016-0203-2

    MATH  Google Scholar 

  33. Xu, X.X., Huang, Q.M., Chen, H.T.: Local superconvergence of continuous Galerkin solutions for delay differential equations of pantograph type. J. Comput. Math. 34, 186–199 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yi, L.J.: An \(L^\infty \)-error estimate for the \(h\)-\(p\) version continuous Petrov–Galerkin method for nonlinear initial value problems. East Asian J. Appl. Math. 5, 301–311 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yi, L.J., Guo, B.Q.: An \(h\)-\(p\) version of the continuous Petrov–Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53, 2677–2704 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yi, L.J., Liang, Z.Q., Wang, Z.Q.: Legendre-Gauss-Lobatto spectral collocation method for nonlinear delay differential equations. Math. Methods Appl. Sci. 36, 2476–2491 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zennaro, M.: Delay differential equations: theory and numerics. In: Ainsworth, M., Levesley, J., Light, W.A., Marietta, M. (eds.) Theory and Numerics of Ordinary and Partial Differential Equations. Clarendon Press, Oxford (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Tingting Meng & Lijun Yi

Authors
  1. Tingting Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijun Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lijun Yi.

Additional information

The work of this author is supported in part by the National Natural Science Foundation of China (Nos. 11301343 and 11571238), the Natural Science Foundation of Shanghai (No. 15ZR1430900), and the Natural Science Foundation of Shanghai Normal University (No. DYL201703).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, T., Yi, L. An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations. J Sci Comput 74, 1091–1114 (2018). https://doi.org/10.1007/s10915-017-0482-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0482-z

Keywords

Mathematics Subject Classification

Search

Navigation