Numerische Mathematik

, Volume 143, Issue 4, pp 797–853

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

Article

## Abstract

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

## References

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)
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)
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)
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)
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)
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)
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)
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)
9. 9.
Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton (1978)
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)
11. 11.
Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)
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)
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)
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)
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)
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)
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)
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)
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)
21. 21.
Dziuk, G., Elliott, C.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)
22. 22.
Dziuk, G., Elliott, C.: $$L^2$$-estimates for the evolving surface finite element method. Math. Comput. 82(281), 1–24 (2013)
23. 23.
Dziuk, G., Kröner, D., Müller, T.: Scalar conservation laws on moving hypersurfaces. Interfaces Free Bound. 15(2), 203–236 (2013)
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)
25. 25.
Ecker, K.: Regularity Theory for Mean Curvature Flow. Springer, Berlin (2012)
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)
27. 27.
Gautschi, W.: Numerical Analysis. Birkhäuser, Boston (1997). An introduction
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)
30. 30.
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
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)
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)
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)
34. 34.
Kovács, B., Lubich, C.: Linearly implicit full discretization of surface evolution. Numer. Math. 140(1), 121–152 (2018)
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)
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)
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)
39. 39.
Persson, P.-O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)
40. 40.
Prohl, A.: Computational Micromagnetism. Teubner, Stuttgart (2001)
41. 41.
Walker, S.W.: The Shape of Things: A Practical Guide to Differential Geometry and the Shape Derivative. SIAM, Philadelphia (2015)
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 