Skip to main content
SpringerLink
Log in

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods.

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

Similar content being viewed by others

References

  1. Akrivis G., Crouzeix M.: Linearly implicit methods for nonlinear parabolic equations. Math. Comput. 73, 613–635 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Akrivis G., Makridakis Ch.: Galerkin time–stepping methods for nonlinear parabolic equations. M2AN Math. Model. Numer. Anal. 38, 261–289 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akrivis G., Makridakis Ch., Nochetto R.H.: A posteriori error estimates for the Crank–Nicolson method for parabolic equations. Math. Comput. 75, 511–531 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Akrivis G., Makridakis Ch., Nochetto R.H.: Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods. Numer. Math. 114, 133–160 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aziz A.K., Monk P.: Continuous finite elements in space and time for the heat equation. Math. Comput. 52, 255–274 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakaev N.Yu.: Linear Discrete Parabolic Problems. North–Holland Mathematical Studies v. 203. Elsevier, Amsterdam (2006)

    Google Scholar 

  7. Brenner P., Crouzeix M., Thomée V.: Single step methods for inhomogeneous linear differential equations in Banach space. RAIRO Anal. Numér. 16, 5–26 (1982)

    MATH  Google Scholar 

  8. Crouzeix, M.: Sur l’approximation des équations différentielles opérationelles linéaires par des méthodes de Runge–Kutta, Thèse, Université de Paris VI (1975)

  9. Eriksson K., Johnson C.: Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28, 43–77 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eriksson K., Johnson C., Larsson S.: Adaptive finite element methods for parabolic problems. VI. Analytic semigroups. SIAM J. Numer. Anal. 35, 1315–1325 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eriksson K., Johnson C., Logg A.: Adaptive computational methods for parabolic problems. In: Stein, E., de Borst, R., Houghes, T.J.R. (eds) Encyclopedia of Computational Mechanics., pp. 1–44. J. Wiley & Sons Ltd., New York (2004)

    Google Scholar 

  12. Estep D., French D.: Global error control for the Continuous Galerkin finite element method for ordinary differential equations. RAIRO Modél. Math. Anal. Numér. 28, 815–852 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Estep D., Larson M., Williams R.: Estimating the error of numerical solutions of systems of reaction-diffusion equations. Mem. Am. Math. Soc. 146(696), 109 (2000)

    MathSciNet  Google Scholar 

  14. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8, Springer, Berlin, 2nd revised ed., 1993, corr. 2nd printing, (2000)

  15. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, Springer, Berlin, 2nd revised ed. (2010)

  16. Lozinski A., Picasso M., Prachittham V.: An anisotropic error estimator for the Crank–Nicolson method: Application to a parabolic problem. SIAM J. Sci. Comp. 31, 2757–2783 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lubich Ch., Ostermann A.: Runge–Kutta methods for parabolic equations and convolution quadrature. Math. Comput. 60, 105–131 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lubich Ch., Ostermann A.: Runge–Kutta approximation of quasi-linear parabolic equations. Math. Comput. 64, 601–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Makridakis Ch., Nochetto R.H.: A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104, 489–514 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nochetto R.H., Schmidt A., Siebert K.G., Veeser A.: Pointwise a posteriori error estimates for monotone semi-linear equations. Numer. Math. 104, 515–538 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nørsett S.P., Wanner G.: Perturbed collocation and Runge–Kutta methods. Numer. Math. 38, 193–208 (1981)

    Article  MathSciNet  Google Scholar 

  22. Thomée V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd ed. Springer, Berlin (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Ioannina, 451 10, Ioannina, Greece

    Georgios Akrivis

  2. Department of Applied Mathematics, University of Crete, 71409, Heraklion-Crete, Greece

    Charalambos Makridakis

  3. Institute of Applied and Computational Mathematics, FORTH, 71110, Heraklion-Crete, Greece

    Charalambos Makridakis

  4. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

Authors
  1. Georgios Akrivis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charalambos Makridakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ricardo H. Nochetto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charalambos Makridakis.

Additional information

Georgios Akrivis was partially supported by the Institute of Applied and Computational Mathematics, FORTH, Greece. Charalambos Makridakis was partially supported by the RTN-network HYKE, HPRN-CT-2002-00282, the EU Marie Curie Dev. Host Site, HPMD-CT-2001-00121 and the program Pythagoras of EPEAEK II. Ricardo H. Nochetto was partially supported by NSF grants DMS-0505454 and DMS-0807811.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akrivis, G., Makridakis, C. & Nochetto, R.H. Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118, 429–456 (2011). https://doi.org/10.1007/s00211-011-0363-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-011-0363-6

Mathematics Subject Classification (2000)

Search

Navigation