Numerische Mathematik

, Volume 143, Issue 4, pp 797–853 | Cite as

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

  • Balázs KovácsEmail author
  • Buyang Li
  • Christian Lubich


A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk’s method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk’s approach in that it discretizes Huisken’s evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

Mathematics Subject Classification

35R01 65M60 65M15 65M12 



We thank Frank Loose and Gerhard Wanner for helpful comments, and we also thank Jörg Nick for helpful discussions. We thank two anonymous referees for their constructive comments on a previous version. The work was partially supported by a grant from the Germany/Hong Kong Joint Research Scheme sponsored by the Research Grants Council of Hong Kong and the German Academic Exchange Service (G-PolyU502/16). The work of Balázs Kovács and Christian Lubich is supported by Deutsche Forschungsgemeinschaft, SFB 1173.


  1. 1.
    Akrivis, G., Li, B., Lubich, C.: Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Mathem. Comput. 86(306), 1527–1552 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barrett, J., 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)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barrett, J., Garcke, H., Nürnberg, R.: On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29(3), 1006–1041 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barrett, J., Garcke, H., Nürnberg, R.: On the parametric finite element approximation of evolving hypersurfaces in \({\mathbb{R}}^3\). J. Comput. Phys. 227(9), 4281–4307 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Barrett, J., Garcke, H., Nürnberg, R.: Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31(1), 225–253 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Barrett, J., Garcke, H., Nürnberg, R.: The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute. Numer. Methods Partial Differ. Equ. 27(1), 1–30 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barrett, J., Garcke, H., Nürnberg, R.: Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves. Numer. Math. 120(3), 489–542 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  10. 10.
    Cimrák, I.: A survey on the numerics and computations for the Landau–Lifshitz equation of micromagnetism. Arch. Comput. Methods Eng. 15(3), 1–37 (2007)MathSciNetGoogle Scholar
  11. 11.
    Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Deckelnick, K.: Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow. Interfaces Free Bound. 2(2), 117–142 (2000)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Deckelnick, K., Dziuk, G.: Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math. 72(2), 197–222 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Deckelnick, K., Dziuk, G.: On the approximation of the curve shortening flow. In: Bandle, C., Chipot, M., Saint Jean Paulin, J., Bemelmans, J., Shafrir, I. (eds.) Calculus of Variations, Applications and Computations, pp. 100–108. Longman Scientific Technical, Harlow (1995)zbMATHGoogle Scholar
  15. 15.
    Deckelnick, K., Dziuk, G.: Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs. Interfaces Free Bound. 2(4), 341–359 (2000)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Malchiodi, A., Neves, A.A., Rivière, T. (eds.) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematis, vol. 1357, pp. 142–155. Springer, Berlin (1988)Google Scholar
  19. 19.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(1), 603–611 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dziuk, G.: Convergence of a semi-discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci. 4(04), 589–606 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dziuk, G., Elliott, C.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dziuk, G., Elliott, C.: \(L^2\)-estimates for the evolving surface finite element method. Math. Comput. 82(281), 1–24 (2013)zbMATHGoogle Scholar
  23. 23.
    Dziuk, G., Kröner, D., Müller, T.: Scalar conservation laws on moving hypersurfaces. Interfaces Free Bound. 15(2), 203–236 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dziuk, G., Lubich, C., Mansour, D.: Runge–Kutta time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 32(2), 394–416 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ecker, K.: Regularity Theory for Mean Curvature Flow. Springer, Berlin (2012)zbMATHGoogle Scholar
  26. 26.
    Elliott, C., Fritz, H.: On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA J. Numer. Anal. 37(2), 543–603 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gautschi, W.: Numerical Analysis. Birkhäuser, Boston (1997). An introductionzbMATHGoogle Scholar
  28. 28.
    Hairer, E., Nø rsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, volume 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993)Google Scholar
  29. 29.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  30. 30.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. In: Hildebrandt, S., Struwe, M. (eds.) Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), volume 1713 of Lecture Notes in Mathematics, pp. 45–84. Springer, Berlin (1999)zbMATHGoogle Scholar
  32. 32.
    Kovács, B.: High-order evolving surface finite element method for parabolic problems on evolving surfaces. IMA J. Numer. Anal. 38(1), 430–459 (2018)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kovács, B., Li, B., Lubich, C., Power Guerra, C.: Convergence of finite elements on an evolving surface driven by diffusion on the surface. Numer. Math. 137(3), 643–689 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kovács, B., Lubich, C.: Linearly implicit full discretization of surface evolution. Numer. Math. 140(1), 121–152 (2018)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Kovács, B., Power Guerra, C.: Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces. Numer. Methods Partial Differ. Equ. 32(4), 1200–1231 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33(4), 1365–1385 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mantegazza, C.: Lecture Notes on Mean Curvature Flow. Progress in Mathematics, vol. 290. Birkhäuser, Basel (2012)Google Scholar
  38. 38.
    Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3(4), 377–423 (1981)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Persson, P.-O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Prohl, A.: Computational Micromagnetism. Teubner, Stuttgart (2001)zbMATHGoogle Scholar
  41. 41.
    Walker, S.W.: The Shape of Things: A Practical Guide to Differential Geometry and the Shape Derivative. SIAM, Philadelphia (2015)zbMATHGoogle Scholar
  42. 42.
    White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002), pp. 525–538. Higher Education Press, Beijing (2002)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Department of Applied MathematicsHong Kong Polytechnic UniversityKowloonHong Kong

Personalised recommendations