An unconditionally stable finite element scheme for anisotropic curve shortening flow

Klaus Deckelnick; Robert Nürnberg

Archivum Mathematicum (2023)

  • Issue: 3, page 263-274
  • ISSN: 0044-8753

Abstract

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Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.

How to cite

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Deckelnick, Klaus, and Nürnberg, Robert. "An unconditionally stable finite element scheme for anisotropic curve shortening flow." Archivum Mathematicum (2023): 263-274. <http://eudml.org/doc/298988>.

@article{Deckelnick2023,
abstract = {Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.},
author = {Deckelnick, Klaus, Nürnberg, Robert},
journal = {Archivum Mathematicum},
keywords = {anisotropic curve shortening flow; finite element method; stability},
language = {eng},
number = {3},
pages = {263-274},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {An unconditionally stable finite element scheme for anisotropic curve shortening flow},
url = {http://eudml.org/doc/298988},
year = {2023},
}

TY - JOUR
AU - Deckelnick, Klaus
AU - Nürnberg, Robert
TI - An unconditionally stable finite element scheme for anisotropic curve shortening flow
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 3
SP - 263
EP - 274
AB - Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.
LA - eng
KW - anisotropic curve shortening flow; finite element method; stability
UR - http://eudml.org/doc/298988
ER -

References

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  1. Alfaro, M., Garcke, H., Hilhorst, D., Matano, H., Schätzle, R., Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation, Proc. Roy. Soc. Edinburgh Sect. A 140 (4) (2010), 673–706. (2010) MR2672065
  2. Barrett, J.W., Blowey, J.F., 10.1007/s002110050276, Numer. Math. 77 (1) (1997), 1–34. (1997) DOI10.1007/s002110050276
  3. Barrett, J.W., Garcke, H., Nürnberg, R., 10.1093/imanum/drm013, IMA J. Numer. Anal. 28 (2) (2008), 292–330. (2008) MR2401200DOI10.1093/imanum/drm013
  4. Barrett, J.W., Garcke, H., Nürnberg, R., 10.1093/imanum/drp005, IMA J. Numer. Anal. 30 (1) (2010), 4–60. (2010) MR2580546DOI10.1093/imanum/drp005
  5. Barrett, J.W., Garcke, H., Nürnberg, R., 10.1002/num.20637, Numer. Methods Partial Differential Equations 27 (1) (2011), 1–30. (2011) MR2743598DOI10.1002/num.20637
  6. Barrett, J.W., Garcke, H., Nürnberg, R., 10.1093/imanum/drt044, IMA J. Numer. Anal. 34 (4) (2014), 1289–1327. (2014) MR3269427DOI10.1093/imanum/drt044
  7. Barrett, J.W., Garcke, H., Nürnberg, R., 10.1093/imanum/drz012, IMA J. Numer. Anal. 40 (3) (2020), 1601–1651. (2020) MR4122486DOI10.1093/imanum/drz012
  8. Barrett, J.W., Garcke, H., Nürnberg, R., Parametric finite element approximations of curvature driven interface evolutions, Handb. Numer. Anal. (Bonito, A., Nochetto, R.H., eds.), vol. 21, Elsevier, Amsterdam, 2020, pp. 275–423. (2020) MR4378429
  9. Bellettini, G., Anisotropic and crystalline mean curvature flow, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, 2004, pp. 49–82. (2004) MR2132657
  10. Bellettini, G., Paolini, M., 10.14492/hokmj/1351516749, Hokkaido Math. J. 25 (3) (1996), 537–566. (1996) Zbl0873.53011DOI10.14492/hokmj/1351516749
  11. Beneš, M., Mikula, K., Simulation of anisotropic motion by mean curvature-comparison of phase field and sharp interface approaches, Acta Math. Univ. Comenian. (N.S.) 67 (1) (1998), 17–42. (1998) 
  12. Caselles, V., Kimmel, R., Sapiro, G., 10.1023/A:1007979827043, Int. J. Comput. Vis. 22 (1) (1997), 61–79. (1997) DOI10.1023/A:1007979827043
  13. Clarenz, U., Dziuk, G., Rumpf, M., On generalized mean curvature flow in surface processing, Geometric Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, 2003, pp. 217–248. (2003) MR2008341
  14. Deckelnick, K., Dziuk, G., On the approximation of the curve shortening flow, Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994) (Bandle, C., Bemelmans, J., Chipot, M., Paulin, J.S.J., Shafrir, I., eds.), Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1995, pp. 100–108. (1995) 
  15. Deckelnick, K., Dziuk, G., Elliott, C.M., 10.1017/S0962492904000224, Acta Numer. 14 (2005), 139–232. (2005) Zbl1113.65097MR2168343DOI10.1017/S0962492904000224
  16. Deckelnick, K., Nürnberg, R., A novel finite element approximation of anisotropic curve shortening flow, arXiv:2110.04605 (2021). 
  17. Deimling, K., Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. (1985) 
  18. Dziuk, G., 10.1142/S0218202594000339, Math. Models Methods Appl. Sci. 4 (4) (1994), 589–606. (1994) DOI10.1142/S0218202594000339
  19. Dziuk, G., 10.1137/S0036142998337533, SIAM J. Numer. Anal. 36 (6) (1999), 1808–1830. (1999) Zbl0942.65112DOI10.1137/S0036142998337533
  20. Ecker, K., Regularity Theory for Mean Curvature Flow, Birkhäuser, Boston, 2004. (2004) Zbl1058.53054MR2024995
  21. Elliott, C.M., 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) (2017), 543–603. (2017) MR3649420
  22. Elliott, C.M., Stuart, A.M., 10.1137/0730084, SIAM J. Numer. Anal. 30 (6) (1993), 1622–1663. (1993) DOI10.1137/0730084
  23. Garcke, H., Lam, K.F., Nürnberg, R., Signori, A., Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies, arXiv:2111.14070 (2021). 
  24. Giga, Y., Surface evolution equations, vol. 99, Birkhäuser, Basel, 2006. (2006) MR2238463
  25. Gurtin, M.E., Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993. (1993) Zbl0787.73004
  26. Haußer, F., Voigt, A., 10.1016/j.aml.2005.05.011, Appl. Math. Lett. 19 (8) (2006), 691–698. (2006) MR2232241DOI10.1016/j.aml.2005.05.011
  27. Mantegazza, C., Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. (2011) MR2815949
  28. Mikula, K., Ševčovič, D., 10.1137/S0036139999359288, SIAM J. Appl. Math. 61 (5) (2001), 1473–1501. (2001) MR1824511DOI10.1137/S0036139999359288
  29. Mikula, K., Ševčovič, D., 10.1002/mma.514, Math. Methods Appl. Sci. 27 (13) (2004), 1545–1565. (2004) MR2077443DOI10.1002/mma.514
  30. Mikula, K., Ševčovič, D., 10.1007/s00791-004-0131-6, Computing Vis. Sci. 6 (4) (2004), 21–225. (2004) MR2071441DOI10.1007/s00791-004-0131-6
  31. Pozzi, P., 10.1002/mma.836, Math. Methods Appl. Sci. 30 (11) (2007), 1243–1281. (2007) MR2334978DOI10.1002/mma.836
  32. Taylor, J.E., Cahn, J.W., Handwerker, C.A., 10.1016/0956-7151(92)90090-2, Acta Metall. Mater. 40 (7) (1992), 1443–1474. (1992) DOI10.1016/0956-7151(92)90090-2
  33. Wu, C., Tai, X., 10.1109/TVCG.2009.103, IEEE Trans. Vis. Comput. Graph. 16 (4) (2010), 647–662. (2010) DOI10.1109/TVCG.2009.103

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