, Volume 53, Issue 1, pp 19–34 | Cite as

On long time integration of the heat equation



We construct space-time Petrov–Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.


Heat equation Long-time Laguerre polynomials Stability 

Mathematics Subject Classification

35K05 65M60 65M12 



The author thanks Ch. Schwab for motivating the topic, and the anonymous referees for their helpful suggestions.


  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions. Dover, New York (10th printing) (1972)Google Scholar
  2. 2.
    Andreev, Roman: Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33(1), 242–260 (2013)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Andreev, Roman: Space-time discretization of the heat equation. Numer. Algorithms 67(4), 713–731 (2014)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Babuška, I.: Error-bounds for finite element method. Numer. Math., 16:322–333 (1970/1971)Google Scholar
  5. 5.
    Babuška, I., Janik, T.: The h-p version of the finite element method for parabolic equations. I. The p version in time. Numer. Methods Partial Differ. Equ. 5, 363–399 (1989)CrossRefMATHGoogle Scholar
  6. 6.
    Bank, Randolph E., Yserentant, Harry: On the \(H^1\)-stability of the \(L_2\)-projection onto finite element spaces. Numer. Math. 126(2), 361–381 (2014)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Courant, R., Hilbert, D.: Methods of mathematical physics. vol. I. Interscience Publishers Inc, New York, 1st English edition, 1953. 7th printing (1966)Google Scholar
  8. 8.
    Gautschi, W.: Orthogonal polynomials. Oxford University Press, Computation and Approximation (2004)Google Scholar
  9. 9.
    Guo, B.-Y., Wang, Z.-Q.: Numerical integration based on Laguerre–Gauss interpolation. Comput. Methods Appl. Mech. Eng. 196(37–40), 3726–3741 (2007)MATHGoogle Scholar
  10. 10.
    Schwab, Christoph, Stevenson, Rob: Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comput. 78(267), 1293–1318 (2009)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Jinchao, Xu, Zikatanov, Ludmil: Some observations on Babuška and Brezzi theories. Numer. Math. 94(1), 195–202 (2003)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.RICAMLinzAustria

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