Computation of geometric partial differential equations and mean curvature flow
Computation of geometric partial differential equations and mean curvature flow
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DOI：
10.1017/s0962492904000224
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发表时间：
2005-04
期刊：
影响因子：
14.2
通讯作者：
K. Deckelnick;G. Dziuk;C. M. Elliott
中科院分区：
文献类型：
--
作者：
K. Deckelnick;G. Dziuk;C. M. Elliott
本文综述了由几何偏微分方程控制的曲率相关界面运动的计算。平均曲率流的规范问题是找到一个曲面，该曲面的演化使得在曲面上的每一点处，法向速度由平均曲率给出。近年来，人们对涉及曲率的几何偏微分方程的兴趣日益浓厚。应用实例包括合金中晶界的运动、相变和图像处理。分析、离散化和数值分析的方法取决于如何表示曲面。最简单的方法是当曲面是基域上的图时。这是一个尖锐的接口方法的例子，在一般的参数化方法，涉及寻求一个参数化的表面上的基础表面，如一个球。另一方面，一个接口可以隐式地表示为一个函数的水平表面，这种想法产生了所谓的水平集方法。另一种隐式方法是相场法，它通过一个满足PDE的相场的零水平集来近似界面，该PDE依赖于一个新的参数。每种方法都有其优点和缺点。在这篇文章中，我们描述了这些方法的数学公式及其离散化。每种方法的算法，收敛结果给出，并支持计算结果和众多的图形。除了平均曲率流，主题的各向异性和高阶几何偏微分方程的Willmore流和表面扩散。
This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.