On the self-similar solutions of the crystalline mean curvature flow in\n three dimensions

2018/06/06 by Norbert Požár, Požár, Norbert
Computer Science · Mathematics · #35K93 #53C44 #65M99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1806.02482

openalex publication_date 2018/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present two types of self-similar shrinking solutions of positive genus\nfor the crystalline mean curvature flow in three dimensions analogous to the\nsolutions known for the standard mean curvature flow. We use them to test a\nnumerical implementation of a level set algorithm for the crystalline mean\ncurvature flow in three dimensions based on the minimizing movements scheme of\nA.~Chambolle, \Interfaces Free Bound.~6~(2004). We implement a finite\nelement method discretization that seems to improve the handling of edges in\nthree dimensions compared to the standard finite difference method and\nillustrate its behavior on a few examples.\n

Citations

• Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
• A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow
• A fast sweeping method for Eikonal equations
• Existence and uniqueness for planar anisotropic and crystalline\n curvature flow

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• Motion by crystalline-like mean curvature: a survey
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• Some algorithms for the mean curvature flow under topological changes