Collaborative Research: Transforming Serendipity Elements from Theory to Practice

该项目旨在显著减少在现代科学和工程应用中进行各种模拟所需的计算时间和工作量。在磁流体动力学（用于核聚变研究）和电扩散（用于心律失常研究）等各种环境中，一种称为“有限元法”的技术是设计和计算相关物理现象的高度精确描述的首选计算程序。虽然有限元方法已经使用了近半个世纪，但最近的数学见解表明，某些类型的方法可以以这样一种方式实现，即它们可以产生相同精度数量级的结果，同时需要的计算量要少几个数量级。这种技术被称为“意外发现方法”，并作为建议工作的重点。这项研究的一个关键影响是将这些方法整合到一个被广泛使用的、开源的、社区开发的软件包中，这个软件包被称为Firedrake项目。从高层次的方法分析到具体的计算技巧，所提出的工作将大大扩展对偶然性有限元方法的优点和局限性的理解。对于意外发现的元素，该项目将通过同时探索通过代数和分析技术的预期计算效益，以及通过Firedrake软件包中的实现和数值模拟的实际计算效益，来实现“相同精度，更少的工作”的共同口号。在理论方面，该研究将通过调查偶然性空间的代数结构及其开发来推进知识，以允许相关多项式空间的新亥姆霍兹分解，识别偶然性基到标准张量积基的有效“短注入”，以及稀疏正交规则的发展。研究人员将结合元素映射和物理定义基元素方面的专业知识，创建非仿射映射方形元素几何的保序处理，这是先前尝试实现意外元素的关键障碍。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to significantly reduce the computation time and effort required to carry out a wide variety of simulations used in modern scientific and engineering applications. In settings as varied as magnetohydrodynamics (used in studies of nuclear fusion) and electro-diffusion (used in studies of cardiac arrhythmia), a technique called the "finite element method" is a preferred computational procedure for designing and computing highly accurate descriptions of the relevant physical phenomena. While finite element methods have been in use for nearly half a century, recent mathematical insights have indicated that certain kinds of methods could be implemented in such a way that they would produce results of the same order of accuracy while requiring orders of magnitude less computational work. Such techniques are known as "serendipity methods" and serve as the focus of the proposed work. A key impact of the research will be incorporation of these methods into a widely used, open source, community-developed software package known as the Firedrake Project.From high-level method analysis to specific computational tricks, the proposed work will significantly expand understanding of the benefits and limitations of serendipity finite element methodologies. For serendipity elements, the project will make precise the common mantra of "same accuracy, less work" by simultaneously exploring the expected computational benefits via algebraic and analytical techniques, as well as actual computational benefits via implementation and numerical simulations within the Firedrake software package. On the theoretical side, the research will advance knowledge via investigation of the algebraic structure of serendipity spaces and its exploitation to allow new Helmholtz decompositions of the associated polynomial spaces, identification of efficient "short injections" of serendipity bases into standard tensor product bases, and development of sparse quadrature rules. The investigators will combine expertise on element mapping and physically-defined basis elements to create an order-preserving handling of non-affinely mapped square element geometries, a key roadblock in prior attempts at implementation of serendipity elements.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.