Finite elements beyond the de Rham complex

在过去的二十年中，有限元外演算（FEEC）已成为现代有限元方法发展的中心。在FEEC框架中，研究的中心对象是Hilbert复形，它是由算子连接的空间链，其中一个算子的值域包含在链中下一个算子的核中。到目前为止，研究得最好的复形是三维的de Rham复形，其中的算子是向量计算的核心格拉德、旋度和div算子。这种复合体对于理解电磁学的麦克斯韦方程和流体力学的斯托克斯方程等是必不可少的。在FEEC中，复形被离散为一个整体;复形中的每个空间都用有限元离散，这样离散空间也用相同的微分算子形成了一个希尔伯特复形。满足这一结构的有限元现在已经被广泛理解，并在开源软件包中广泛使用。许多其他复形是物理学其他领域的核心;例如，弹性复形支撑着固体力学，海森复形支撑着广义相对论的爱因斯坦方程。然而，德拉姆复形之外的复形的离散化却知之甚少。只有部分知识的有限元素，他们是可用的，和由此产生的有限元素是外来的和unavailable在任何强大的mathematicalsoftware.In这个项目中，我们将开发数学和计算工具的离散化的复杂的de Rham复杂。这项工作将位于数值分析和计算机科学之间的接口。该项目的第一步将是对目前已知的弹性和海森复体的离散化进行彻底的审查。利用这些知识，我们将制定一项战略，使新的有限元快速实施。我们预计这将涉及在Firedrake软件中设计一种特定于领域的语言，以允许代码中的有限元素的软件规范与数学规范同构。我们将验证我们开发的领域特定语言，方法是使用它来实现已经手工实现的外来元素（如Argyris和Arnold-Winther元素），然后超越这些元素，再使用尚未实现的元素（如Zhang、Bogner-Fox-施密特和Hu-Zhang-Zhang元素）。一旦实现了核心语言，我们将研究离散Bernstein-Gelfand-Gelfand（BGG）构造，这是一个重要的数学扩展。BGG构造从德拉姆复形的几个副本中导出（连续）弹性、海森复形和其他复形。正如BGG结构可以应用于连续水平，我们推测，它也可以应用于离散水平，揭示了如何从我们的知识，如何离散的de Rham复杂的其他复杂的洞察力。这项研究将使广泛的偏微分方程的数值解，目前是棘手的或需要严重缺陷的离散，包括电磁学中的高阶卷曲-卷曲问题、近晶A型液晶的Pevnyi-Selinger-Sluckin模型和超弹性固体材料的应力-位移公式，该项目属于EPSRC数值分析研究领域的福尔斯。

Over the past twenty years the finite element exterior calculus (FEEC) has become central to the modern development of the finite element method. In the FEEC frame-work, the central object of study is a Hilbert complex, which is a chain of spaces linked by operators, with the range of one operator being contained in the kernel of the next in the chain. By far the best-studied complex is the so-called de Rham complex in three dimen-sions, in which the operators are the core grad, curl, and div operators of vector cal-culus. This complex is essential to the understanding of the Maxwell equations of electromagnetics and the Stokes equations of fluid mechanics, among others. In FEEC, the complex is discretised as a whole; each space in the complex is discre-tised with a finite element, so that the discrete spaces also form a Hilbert complex in their own right with the same differential operators. Finite elements satisfying this structure are now well-understood and widely available in open source software packages.Many other complexes are core to other areas of physics; for example, the elasticity complex underpins solid mechanics, and the Hessian complex underpins the Einstein equations of general relativity. However, the discretisation of complexes beyond the de Rham complex is very poorly understood. Only partial knowledge of finite ele-ments for them is available, and the resulting finite elements are exotic and unavaila-ble in any robust mathematical software.In this project we will develop mathematical and computational tools for the discreti-sation of complexes beyond the de Rham complex. This work will lie at the interface between numerical analysis and computer science .The first step in the project will be to conduct a thorough review of what is currently known about the discretisation of the elasticity and Hessian complexes. Using this knowledge, we will then develop a strategy that will enable the rapid implementation of new finite elements. We anticipate that this will involve designing a domain-specific language in the Firedrake software to allow the software specification of finite ele-ments in code that is isomorphic to the mathematical specification. We will verify the domain-specific language we develop by using it to implement exotic elements that are already implemented by hand (such as the Argyris and Arnold-Winther elements) and then move beyond these to elements that are not yet available (such as the Zhang, Bogner-Fox-Schmidt, and Hu-Zhang-Zhang elements). Once the core language is implemented, we will investigate the discrete Bernstein-Gelfand-Gelfand (BGG) construction, an important mathematical extension. The BGG construction derives the (continuous) elasticity, Hessian, and other complexes from several copies of the de Rham complex. Just as the BGG construction can be applied at the continuous level, we conjecture that it can also be applied at the discrete level, revealing insight into how to discretise other complexes from our knowledge of how to discretise the de Rham complex.This research will enable the numerical solution of a wide range of partial differential equations that are currently intractable or require nonconforming discretisations with serious drawbacks, including high-order curl-curl problems in electromagnetics, the Pevnyi-Selinger-Sluckin model of smectic-A liquid crystals, and the stress-displacement formulation of hyperelastic solid materials.This project falls within the EPSRC Numerical Analysis research area.