Numerische Mathematik

, Volume 140, Issue 1, pp 121–152 | Cite as

Linearly implicit full discretization of surface evolution

  • Balázs Kovács
  • Christian Lubich


Stability and convergence of full discretizations of various surface evolution equations are studied in this paper. The proposed discretization combines a higher-order evolving-surface finite element method for space discretization with higher-order linearly implicit backward difference formulae for time discretization. The stability of the full discretization is studied in the matrix–vector formulation of the numerical method. The geometry of the problem enters into the bounds of the consistency errors, but does not enter into the proof of stability. Numerical examples illustrate the convergence behaviour of the full discretization.

Mathematics Subject Classification

35R01 65M60 65M15 65M12 



This work is supported by Deutsche Forschungsgemeinschaft, SFB 1173.


  1. 1.
    Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131(4), 713–735 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akrivis, G., Li, B., Lubich, C.: Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comput. 86(306), 1527–1552 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barrett, J.W., Deckelnick, K., Styles, V.: Numerical analysis for a system coupling curve evolution to reaction diffusion on the curve. SIAM J. Numer. Anal. 55(2), 1080–1100 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demlow, A.: Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47(2), 805–807 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol. 1357, pp. 142–155. Springer, Berlin (1988)Google Scholar
  7. 7.
    Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dziuk, G., Elliott, C.M.: Fully discrete evolving surface finite element method. SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dziuk, G., Elliott, C.M.: \(L^2\)-estimates for the evolving surface finite element method. Math. Comput. 82(281), 1–24 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dziuk, G., Lubich, C., Mansour, D.E.: Runge-Kutta time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 32(2), 394–416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gautschi, W.: Numerical Analysis, 1st edn. Birkauser, Boston (1997)zbMATHGoogle Scholar
  12. 12.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differetial-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kovács, B.: High-order evolving surface finite element method for parabolic problems on evolving surfaces. IMA J. Numer. Anal. 38(1), 430–459 (2018). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kovács, B., Li, B., Lubich, C., Power Guerra, C.A.: Convergence of finite elements on an evolving surface driven by diffusion on the surface. Numer. Math. 137(3), 643–689 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kovács, B., Power Guerra, C.A.: Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces. Numer. Methods Partial Differ. Equ. 32(4), 1200–1231 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lubich, C., Mansour, D.E., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33(4), 1365–1385 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3(4), 377–423 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

Personalised recommendations