Phase field method for mean curvature flow with boundary constraints

Bretin, Elie; Perrier, Valerie

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 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.

Classification : 49Q, 35B, 35K
Mots clés : Allen Cahn equation, mean curvature flow, boundary constraints, penalization technique, gamma-convergence, Fourier splitting method

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