Free Access
Issue |
ESAIM: M2AN
Volume 46, Number 6, November-December 2012
|
|
---|---|---|
Page(s) | 1509 - 1526 | |
DOI | https://doi.org/10.1051/m2an/2012014 | |
Published online | 13 June 2012 |
- 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. [Google Scholar]
- 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. [CrossRef] [Google Scholar]
- L. Almeida, A. Chambolle and M. Novaga, Mean curvature flow with obstacle. Technical Report Preprint (2011). [Google Scholar]
- 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. [Google Scholar]
- G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris 17 (1994). [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Convex Anal. 4 (1997) 91–108. [Google Scholar]
- 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. [Google Scholar]
- G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537–566. [MathSciNet] [Google Scholar]
- B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization. ESAIM : COCV 9 (2003) 19–48. [Google Scholar]
- M. Brassel, Instabilité de Forme en Croissance Cristalline. Ph.D. thesis, University Joseph Fourier, Grenoble (2008). [Google Scholar]
- 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. [Google Scholar]
- E. Bretin, Méthode de champ de phase et mouvement par courbure moyenne. Ph.D. thesis, Institut National Polytechnique de Grenoble (2009). [Google Scholar]
- 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. [CrossRef] [Google Scholar]
- X. Chen, Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equ. 96 (1992) 116–141. [CrossRef] [MathSciNet] [Google Scholar]
- L.Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108 (1998) 147–158. [CrossRef] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- K. Deckelnick and G. Dziuk, Discrete anisotropic curvature flow of graphs. ESAIM : M2AN 33 (1999) 1203–1222. [CrossRef] [EDP Sciences] [Google Scholar]
- K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. [CrossRef] [MathSciNet] [Google Scholar]
- L.C. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differ. Geom. 33 (1991) 635–681. [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- 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. [CrossRef] [Google Scholar]
- 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. [CrossRef] [Google Scholar]
- 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. [MathSciNet] [Google Scholar]
- L. Modica and S. Mortola, Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. [MathSciNet] [Google Scholar]
- S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York. Appl. Math. Sci. (2002). [Google Scholar]
- S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision and Graphics. Springer-Verlag, New York (2003). [Google Scholar]
- 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. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- M. Röger and R. Schätzle, On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675–714. [CrossRef] [MathSciNet] [Google Scholar]
- R. Schätzle, Lower semicontinuity of the Willmore functional for currents. J. Differ. Geom. 81 (2009) 437–456. [Google Scholar]
- R. Schätzle, The Willmore boundary problem. Calc. Var. Partial Differ. Equ. 37 (2010) 275–302. [CrossRef] [Google Scholar]
- S. Serfaty, Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Disc. Cont. Dyn. Systems 31 (2011) 1427–1451. [Google Scholar]
- 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. [Google Scholar]
- 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. [CrossRef] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.