Geodesic
Phase field method for mean curvature flow with boundary constraints
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1509-1526

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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 : 10.1051/m2an/2012014
Classification : 49Q, 35B, 35K
Keywords: Allen Cahn equation, mean curvature flow, boundary constraints, penalization technique, gamma-convergence, Fourier splitting method 
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     author = {Bretin, Elie and Perrier, Valerie},
     title = {Phase field method for mean curvature flow with boundary constraints},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1509--1526},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     doi = {10.1051/m2an/2012014},
     mrnumber = {2996338},
     zbl = {1272.65057},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012014/}
}
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Bretin, Elie; Perrier, Valerie. Phase field method for mean curvature flow with boundary constraints. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1509-1526. doi: 10.1051/m2an/2012014

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