同调调和余链
Cohomologous Harmonic Cochains
AI总结：
本文提出算法寻找调和余链，用于外微积分中求解椭圆偏微分方程，并通过离散Hodge-deRham定理导出加权最小二乘法，实现与给定上闭链同调的调和余链的局部化。
摘要：
我们描述了寻找调和余链的算法，这是在外微积分中求解椭圆偏微分方程的关键要素。调和余链在计算拓扑和计算机图形学中也很有用。我们专注于寻找与给定上闭链同调的调和余链。除此之外，这允许在感兴趣拓扑特征附近进行局部化。我们通过证明调和余链空间与上同调空间之间的离散Hodge-deRham定理，推导出一种加权最小二乘法。根据所使用的离散Hodge星算子，所得到的解要么满足Whitney形式有限元外微积分方程，要么满足调和余链的离散外微积分方程。
英文摘要：
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows localization near topological features of interest. We derive a weighted least squares method by proving a discrete Hodge-deRham theorem on the isomorphism between the space of harmonic cochains and cohomology. The solution obtained either satisfies the Whitney form finite element exterior calculus equations or the discrete exterior calculus equations for harmonic cochains, depending on the discrete Hodge star used.
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