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/

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