[论文解读] Discrete exterior calculus

Jerrold E. Marsden, Anil N. Hirani|arXiv (Cornell University)|Jan 1, 2003

Algebraic and Geometric Analysis参考文献 57被引用 490

一句话总结

本文提出了一种仅基于单纯复形及其对偶上的离散组合与几何运算的离散外微分形式（DEC）框架，引入了离散微分形式、向量场及与连续对应物相匹配的算子。其关键贡献在于通过受控插值实现纯离散化表述，能够精确恢复已知公式，并为变分问题与约束系统中的结构保持型数值方法提供基础。

ABSTRACT

This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex and its geometric dual. The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. Definitions are given for discrete versions of all the usual operators of exterior calculus. The presence of forms and vector fields allows us to address their various interactions, which are important in applications. In many examples we find that the formulas derived from DEC are identical to the existing formulas in the literature. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. One potential application of DEC is to variational problems which come equipped with a rich exterior calculus structure. On the discrete level, such structures will be enhanced by the availability of DEC. One of the objectives of this thesis is to fill this gap. There are many constraints in numerical algorithms that naturally involve differential forms. Preserving such features directly on the discrete level is another goal, overlapping with our goals for variational problems. In this thesis we have tried to push a purely discrete point of view as far as possible. We argue that this can only be pushed so far, and that interpolation is a useful device. For example, we found that interpolation of functions and vector fields is a very convenient. In future work we intend to continue this interpolation point of view, extending it to higher degree forms, especially in the context of the sharp, Lie derivative and interior product operators. Some preliminary ideas on this point of view are presented in the thesis. We also present some preliminary calculations of formulas on regular nonsimplicial complexes.

研究动机与目标

• 基于单纯复形及其对偶上的离散几何与组合运算，建立纯离散的离散外微分形式（DEC）框架。
• 在离散意义上定义微分形式、向量场及所有标准外微分形式算子的离散类比，包括外微分、共微分与霍奇星算子。
• 证明单纯复形的外接圆心对偶复形自然支持离散理论中的度量相关运算。
• 在数值算法中保持内在的几何与拓扑结构，尤其适用于涉及微分形式与变分原理的问题。
• 探讨插值在将DEC扩展至高阶形式以及高级算子（如尖点算子、李导数与内乘算子）中的作用。

提出的方法

• 理论构建仅依赖于单纯复形及其几何对偶上的离散数据，避免使用连续流形。
• 离散微分形式通过单纯复形上的余链复形定义，而离散向量场则在对偶复形上定义。
• 外微分与共微分等算子通过组合方式定义，霍奇星算子则由外接圆心对偶导出。
• 引入函数与向量场的插值作为工具，以在保持与连续理论一致性的前提下扩展离散框架。
• 通过证明离散公式能重现文献中的已知结果，验证了该框架的有效性。
• 利用几何与组合推理，初步探讨了在规则非单纯复形上的扩展。

实验结果

研究问题

• RQ1 如何仅通过单纯复形及其对偶上的离散几何与组合运算，构建离散外微分形式？
• RQ2 外接圆心对偶在实现离散外微分形式中度量相关运算方面起到什么作用？
• RQ3 在完全离散的设定下，如何一致地定义离散向量场及其与微分形式的相互作用？
• RQ4 函数与向量场的插值在不依赖连续流形的前提下，以何种方式增强离散微分形式框架？
• RQ5 离散算子与结构能否恢复经典外微分形式中的已知公式？在何种条件下可以实现？

主要发现

• 当应用于标准例子时，该离散外微分形式框架成功重现了文献中的已知公式，证实了其与连续理论的一致性。
• 单纯复形的外接圆心对偶为定义如霍奇星算子等度量相关算子提供了自然且有效的结构。
• 离散形式与向量场可分别在单纯复形及其对偶上定义，从而在离散设定下研究其相互作用成为可能。
• 函数与向量场的插值是一种有用且便捷的工具，可在不依赖连续流形的前提下增强离散框架。
• 该理论支持丰富的代数与几何结构，包括尖点算子、李导数与内乘算子的离散类比，初步提出了未来扩展的思路。
• 在规则非单纯复形上的初步结果表明，基于几何与组合原理，DEC框架可推广至非单纯网格。