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.
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 },
pages = {1509--1526},
year = {2012},
publisher = {EDP Sciences},
volume = {46},
number = {6},
doi = {10.1051/m2an/2012014},
mrnumber = {2996338},
zbl = {1272.65057},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2012014/}
}
TY - JOUR AU - Bretin, Elie AU - Perrier, Valerie TI - Phase field method for mean curvature flow with boundary constraints JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1509 EP - 1526 VL - 46 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2012014/ DO - 10.1051/m2an/2012014 LA - en ID - M2AN_2012__46_6_1509_0 ER -
%0 Journal Article %A Bretin, Elie %A Perrier, Valerie %T Phase field method for mean curvature flow with boundary constraints %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1509-1526 %V 46 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2012014/ %R 10.1051/m2an/2012014 %G en %F 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 , Tome 46 (2012) no. 6, pp. 1509-1526. doi: 10.1051/m2an/2012014
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