Anisotropic motion by mean curvature in the context of Finsler geometry

Bellettini, Giovanni; Paolini, Maurizio

doi:10.14492/hokmj/1351516749

Cited by 190 publications

(183 citation statements)

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“…We refer to [4,11,24] for further information and conclude this section by two examples of anisotropy functions γ satisfying our assumptions (A1) and (A2).…”

Section: Continuous Problemmentioning

confidence: 98%

“…Anisotropic mean curvature flow of a surface Γ with normal n is characterized by the steepest descent of the anisotropic surface energy Γ γ(n) dΓ with respect to the Finsler metric associated with the anisotropy function γ [4]. Note that classical mean curvature flow is recovered for the Euclidean distance γ = | · |.…”

Section: Introductionmentioning

confidence: 99%

“…In their pioneering paper, Bellettini and Paolini [4] used formal asymptotics to show that anisotropic mean curvature flow can be approximated by anisotropic versions of the Allen-Cahn equation, as obtained by L 2 gradient flow of the anisotropic Ginzburg-Landau energy…”

Section: Introductionmentioning

confidence: 99%

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Time discretizations of anisotropic Allen-Cahn equations

Gräser

Kornhuber

Sack

2013

IMA Journal of Numerical Analysis

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Abstract. We consider anisotropic Allen-Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semi-implicit time discretizations of the double-well potential. In the semi-implicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer frame invariant, but depends on the orientation of the initial configuration.

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“…We refer to [4,11,24] for further information and conclude this section by two examples of anisotropy functions γ satisfying our assumptions (A1) and (A2).…”

Section: Continuous Problemmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Time discretizations of anisotropic Allen-Cahn equations

Gräser

Kornhuber

Sack

2013

IMA Journal of Numerical Analysis

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“…The unit normal vector ν to Γ t is given by ν = (∇u, −1) 1 + |∇u| 2 while the normal velocity is computed as…”

Section: γ T = {(X U(x T)) | X ∈ ω}·mentioning

confidence: 99%

Discrete anisotropic curvature flow of graphs

Deckelnick¹,

Dziuk²

1999

ESAIM: M2AN

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Abstract. The evolution of n-dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfaces in natural norms. Furthermore we prove a global existence result for the continuous problem and present some examples of computed surfaces.Résumé. L'évolution de graphes n-dimensionnels selon le problème de flotà courbure pondérée est approchée par deséléments finis linéaires. On obtient des bornes d'erreurs optimales pour les normales et pour les vitesses normales des surfaces, dans des normes naturelles. De plus, nousétablissons un théorème d'existence globale pour le problème continu et nous présentons quelques exemples numériques d'évolution de surfaces.

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“…These can be roughly classified into three categories: parametric methods [6,7,22,23], Level set formulations [19,24,[32][33][34] or Phase field approaches [11,17,31,36]. See for instance [23] for a complete review and comparison beetween these three differents strategies.…”

Section: Introductionmentioning

confidence: 99%