### Abstract

Motion by (weighted) mean curvature is a geometric evolution law forsurfaces, representing steepest descent with respect to (an)isotropicsurface energy. It has been proposed that this motion couldbe computed by solving the analogous evolution law using a``crystalline'' approximation to the surface energy. We present thefirst convergence analysis for a numerical scheme of this type. Ourtreatment is restricted to one dimensional surfaces (curves in theplane) which are graphs. In this context, the scheme amounts to a newalgorithm for solving quasilinear parabolic equations in one spacedimension.

Original language | English (US) |
---|---|

Pages (from-to) | 41-70 |

Number of pages | 30 |

Journal | Numerische Mathematik |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - 1994 |

### Fingerprint

### Keywords

- Mathematics Subject Classification (1991): 65M12, 73B30, 35K20

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

**Convergence of a crystalline algorithmfor the heat equation in one dimensionand for the motion of a graphby weighted curvature.** / Gir\~ao, Pedro Martins; Kohn, Robert.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 67, no. 1, pp. 41-70. https://doi.org/10.1007/s002110050017

}

TY - JOUR

T1 - Convergence of a crystalline algorithmfor the heat equation in one dimensionand for the motion of a graphby weighted curvature

AU - Gir\~ao, Pedro Martins

AU - Kohn, Robert

PY - 1994

Y1 - 1994

N2 - Motion by (weighted) mean curvature is a geometric evolution law forsurfaces, representing steepest descent with respect to (an)isotropicsurface energy. It has been proposed that this motion couldbe computed by solving the analogous evolution law using a``crystalline'' approximation to the surface energy. We present thefirst convergence analysis for a numerical scheme of this type. Ourtreatment is restricted to one dimensional surfaces (curves in theplane) which are graphs. In this context, the scheme amounts to a newalgorithm for solving quasilinear parabolic equations in one spacedimension.

AB - Motion by (weighted) mean curvature is a geometric evolution law forsurfaces, representing steepest descent with respect to (an)isotropicsurface energy. It has been proposed that this motion couldbe computed by solving the analogous evolution law using a``crystalline'' approximation to the surface energy. We present thefirst convergence analysis for a numerical scheme of this type. Ourtreatment is restricted to one dimensional surfaces (curves in theplane) which are graphs. In this context, the scheme amounts to a newalgorithm for solving quasilinear parabolic equations in one spacedimension.

KW - Mathematics Subject Classification (1991): 65M12, 73B30, 35K20

UR - http://www.scopus.com/inward/record.url?scp=84951613405&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951613405&partnerID=8YFLogxK

U2 - 10.1007/s002110050017

DO - 10.1007/s002110050017

M3 - Article

VL - 67

SP - 41

EP - 70

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 1

ER -