Phase field method for mean curvature flow with boundary constraints
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1509-1526

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models.

DOI : https://doi.org/10.1051/m2an/2012014
Classification:  49Q,  35B,  35K
Keywords: Allen Cahn equation, mean curvature flow, boundary constraints, penalization technique, gamma-convergence, Fourier splitting method
@article{M2AN_2012__46_6_1509_0,
author = {Bretin, Elie and Perrier, Valerie},
title = {Phase field method for mean curvature flow with boundary constraints},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
pages = {1509-1526},
doi = {10.1051/m2an/2012014},
zbl = {1272.65057},
mrnumber = {2996338},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_6_1509_0}
}

Bretin, Elie; Perrier, Valerie. Phase field method for mean curvature flow with boundary constraints. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, pp. 1509-1526. doi : 10.1051/m2an/2012014. http://www.numdam.org/item/M2AN_2012__46_6_1509_0/

[1] G. Alberti, Variational models for phase transitions, an approach via γ-convergence, in Calculus of variations and partial differential equations (Pisa, 1996). Springer, Berlin (2000) 95-114. | MR 1757697 | Zbl 0957.35017

[2] S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085-1095.

[3] L. Almeida, A. Chambolle and M. Novaga, Mean curvature flow with obstacle. Technical Report Preprint (2011). | Numdam | MR 2971026 | Zbl 1252.49072

[4] L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, in Calculus of variations and partial differential equations (Pisa, 1996). Springer, Berlin (2000) 5-93. | MR 1757696 | Zbl 0956.35002

[5] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris 17 (1994). | MR 1613876 | Zbl 0819.35002

[6] J.W. Barrett, H. Garcke and R. Nürnberg, On the parametric finite element approximation of evolving hypersurfaces in r3. J. Comput. Phys. 227 (2008) 4281-4307. | MR 2406538 | Zbl 1145.65068

[7] J.W. Barrett, H. Garcke and R. Nürnberg, A variational formulation of anisotropic geometric evolution equations in higher dimensions. Numer. Math. 109 (2008) 1-44. | MR 2377611 | Zbl 1149.65082

[8] P.W. Bates, S. Brown and J.L. Han, Numerical analysis for a nonlocal Allen-Cahn equation. Int. J. Numer. Anal. Model. 6 (2009) 33-49. | MR 2574896 | Zbl 1165.65051

[9] G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Convex Anal. 4 (1997) 91-108. | MR 1459883 | Zbl 0882.49013

[10] G. Bellettini and M. Paolini, Quasi-optimal error estimates for the mean curvature flow with a forcing term. Differ. Integral Equ. 8 (1995) 735-752. | MR 1306590 | Zbl 0820.49019

[11] G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537-566. | MR 1416006 | Zbl 0873.53011

[12] B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization. ESAIM : COCV 9 (2003) 19-48. | Numdam | MR 1957089 | Zbl 1066.49029

[13] M. Brassel, Instabilité de Forme en Croissance Cristalline. Ph.D. thesis, University Joseph Fourier, Grenoble (2008).

[14] M. Brassel and E. Bretin, A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 34 (2011) 1157-1180. | MR 2838769 | Zbl 1235.49082

[15] E. Bretin, Méthode de champ de phase et mouvement par courbure moyenne. Ph.D. thesis, Institut National Polytechnique de Grenoble (2009).

[16] A. Bueno-Orovio, V.M. Pérez-García and F.H. Fenton, Spectral methods for partial differential equations in irregular domains : The spectral smoothed boundary method. SIAM J. Sci. Comput. 28 (2006) 886-900. | MR 2240795 | Zbl 1114.65119

[17] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equ. 96 (1992) 116-141. | MR 1153311 | Zbl 0765.35024

[18] L.Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108 (1998) 147-158. | Zbl 1017.65533

[19] Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Proc. Jpn Acad. Ser. A 65 (1989) 207-210. | MR 1030181 | Zbl 0735.35082

[20] X.F. Chen, C.M. Elliott, A. Gardiner and J.J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation. Appl. Anal. 69 (1998) 47-56. | MR 1708186 | Zbl 0992.65096

[21] M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull Amer. Math. Soc. 27 (1992) 1-68. | MR 1118699 | Zbl 0755.35015

[22] K. Deckelnick and G. Dziuk, Discrete anisotropic curvature flow of graphs. ESAIM : M2AN 33 (1999) 1203-1222. | Numdam | MR 1736896 | Zbl 0948.65138

[23] K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139-232. | MR 2168343 | Zbl 1113.65097

[24] L.C. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differ. Geom. 33 (1991) 635-681. | MR 1100206 | Zbl 0726.53029

[25] L.C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45 (1992) 1097-1123. | MR 1177477 | Zbl 0801.35045

[26] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33-65. | MR 1971212 | Zbl 1029.65093

[27] X. Feng and A. Prohl, Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73 (2004) 541-567. | MR 2028419 | Zbl 1115.76049

[28] X. Feng and H.-J. Wu, A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow. J. Sci. Comput. 24 (2005) 121-146. | MR 2221163 | Zbl 1096.76025

[29] Y. Li, H.G. Lee, D. Jeong and J. Kim, An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl. 60 (2010) 1591-1606. | MR 2679126 | Zbl 1202.65112

[30] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14 (1977) 526-529. | MR 473971 | Zbl 0364.49006

[31] L. Modica and S. Mortola, Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR 445362 | Zbl 0356.49008

[32] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York. Appl. Math. Sci. (2002). | MR 1939127 | Zbl 1026.76001

[33] S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision and Graphics. Springer-Verlag, New York (2003). | MR 2071628 | Zbl 1027.68137

[34] S. Osher and J.A. Sethian, Fronts propagating with curvature-dependent speed : algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | MR 965860 | Zbl 0659.65132

[35] N.C. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a dirichlet condition. Proc. R. Soc. London 429 (1990) 505-532. | MR 1057968 | Zbl 0722.49021

[36] M. Paolini, An efficient algorithm for computing anisotropic evolution by mean curvature, in Curvature flows and related topics, edited by Levico, 1994. Gakuto Int. Ser. Math. Sci. Appl. 5 (1995) 199-213. | MR 1365309 | Zbl 0838.73079

[37] M. Röger and R. Schätzle, On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675-714. | Zbl 1126.49010

[38] R. Schätzle, Lower semicontinuity of the Willmore functional for currents. J. Differ. Geom. 81 (2009) 437-456. | MR 2472179 | Zbl 1214.53011

[39] R. Schätzle, The Willmore boundary problem. Calc. Var. Partial Differ. Equ. 37 (2010) 275-302. | MR 2592972 | Zbl 1188.53006

[40] S. Serfaty, Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Disc. Cont. Dyn. Systems 31 (2011) 1427-1451. | MR 2836361 | Zbl 1239.35015

[41] H.-C.Y. Yu, H.-Y. Chen and K. Thornton, Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries. Technical Report, arXiv:1107.5341v1 (2011). Submitted.

[42] J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31 (2009) 3042-3063. | MR 2520311 | Zbl 1198.82045