JournalsifbVol. 14, No. 1pp. 1–35

Consistency result for a non monotone scheme for anisotropic mean curvature flow

  • Eric Bonnetier

    Université Joseph Fourier, Grenoble, France
  • Elie Bretin

    INSA de Lyon, Villeurbanne, France
  • Antonin Chambolle

    Ecole Polytechnique, Palaiseau, France
Consistency result for a non monotone scheme for anisotropic mean curvature flow cover
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Abstract

In this paper, we propose a new scheme for anisotropic motion by mean curvature in Rd\mathbb R^d. The scheme consists of a phase-field approximation of the motion, where the nonlinear diffusive terms in the corresponding anisotropic Allen-Cahn equation are linearized in the Fourier space. In real space, this corresponds to the convolution with a specific kernel of the form

Kϕ,t(x)=F1[e4π2tϕo(ξ)](x).K_{\phi,t}(x) = \mathcal F^{-1}\left[ e^{-4\pi^2 t \phi^o(\xi)} \right](x).

We analyse the resulting scheme, following the work of Ishii-Pires-Souganidis on the convergence of the Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main difficulty here, is that the kernel Kϕ,tK_{\phi,t} is not positive and that its moments of order 2 are not in L1(Rd)L^1(\mathbb R^d). Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow.

Cite this article

Eric Bonnetier, Elie Bretin, Antonin Chambolle, Consistency result for a non monotone scheme for anisotropic mean curvature flow. Interfaces Free Bound. 14 (2012), no. 1, pp. 1–35

DOI 10.4171/IFB/272