Subdivision and the Construction of Smooth Bases for Discrete Differential Forms

为了模拟物理现象，如流体流动或电磁学，必须将基础方程映射为可以在计算机上有效可靠地执行的算法。这种映射过程涉及离散化，例如，将光滑表面映射到三角形网格。为了确保结果计算仍然有意义，必须执行离散化步骤，以便尽可能多地保留重要的物理结构。在数学术语中，我们必须找到微分和积分运算符的离散类似物，它们遵循与抽象数学对应符相同的定理和结构。为此，PI和他的学生将带来从几何建模（细分方案）到离散基形式的构建（这是合适的数值算法的基础）的工具。这样的工具将使在感兴趣的物理现象的核心方程的更高阶和更精确的模拟成为可能。为数值模拟建立这样的基础算法将提高它们的可靠性和速度，并扩大它们在许多科学和工程学科中的影响。当离散偏微分方程用于数值模拟时，数值精度、稳定性和收敛性是重点。直到最近几年，人们才充分认识到，保留连续结构，如对称群及其相关动量，对数值模拟的性能也有同样巨大的影响，在某些情况下，似乎对完全预测模拟至关重要。离散外微积分作为研究积分和微分基本算子的离散实现的框架而出现。到目前为止，潜在的理论本质上是一个局部逼近，没有伴随的全局光滑性（事实上，它的大部分都是分段线性的）。本研究利用可细化函数构造中已知的工具，在简单三流形（带边界）上构造离散微分k型的光滑基。特别地，下面的deRham复合体是由简单共边界算子推导出来的。这种类型的分段线性形式在经典中是已知的。该研究小组最近证明了可细化性的概念与某些离散交换关系足以在简单的两个流形设置中产生更高光滑性的基。三流形的情况（使用四面体而不是三角形）相当具有挑战性，因为不存在三个空间的正四面体化。此外，结果结构的平滑性分析将需要新的工具，因为需要检查新的奇异情况类别，并且违反了可细化函数理论的标准假设。研究人员将构建（在提供算法及其实现的意义上）并分析这些结构。

In order to simulate physical phenomena, such as fluid flow or electromagnetism, one must map the underlying equations to algorithms that can execute efficiently and reliably on a computer. This process of mapping involves discretization, for example, mapping a smooth surface to a triangle mesh. To ensure that the resulting computations are still meaningful this discretization step must be performed so as to preserve as many of the important physical structures as possible. In mathematical terms one must find discrete analogs for the operators of differential and integral calculus which follow the same theorems and structures as their abstract mathematical counterparts. To this end the PI together with his students will bring tools from geometric modeling (subdivision schemes) to the construction of discrete basis forms (which are the foundation of suitable numerical algorithms). Such tools will enable higher order and more accurate simulation of the equations at the heart of physical phenomena of interest. Building such foundational algorithms for numerical simulation will increase their reliability and speed and extend their impact in numerous scientific and engineering disciplines. When discretizing partial differential equations for purposes of numerical simulation much emphasis has been placed on numerical accuracy, stability, and convergence. Only in recent years has it been fully appreciated that preservation of continuous structures such as symmetry groups and their associated momenta can have equally dramatic impact on the performance of numerical simulations, and in some cases appears to be essential to fully predictive simulations. Discrete Exterior Calculus has emerged as a framework in which to study such discrete realizations of the basic operators of integral and differential calculus. So far the underlying theory has essentially been one of local approximation without attendant global smoothness properties (in fact, much of it has been piecewise linear only). This research uses tools known from the construction of refinable functions to construct smooth bases of discrete differential k-forms on simplicial three manifolds (with boundary). In particular the underlying deRham complex is induced by the simplicial co-boundary operator. Piecewise linear forms of this type have been known classically. The research team recently demonstrated that the concept of refinability together with certain discrete commutative relations are sufficient to produce bases of higher smoothness in the simplicial two manifold setting. The three manifold case (using tetrahedra rather than triangles) is considerably more challenging since no regular tetrahedralizations of three space exist. Furthermore the smoothness analysis of the resulting constructions will require new tools since new classes of singular cases need to be examined and standard assumption of refinable function theory are violated. The researchers will both construct (in the sense of providing algorithms and their implementation) and analyze such constructions.