Finite elements beyond the de Rham complex

在过去的二十年里，有限元外积分(FEEC)已经成为现代有限元方法发展的核心。在FEEC框架中，中心研究对象是希尔伯特复合体，它是由算子连接的空间链，一个算子的范围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所谓的三维德罗姆复形，其中的算符是向量微积分的核心梯度、旋度和除法算子。这种复杂性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，这样离散的空间也可以用相同的微分算子形成一个独立的希尔伯特复形。满足这种结构的有限元现在已经很好地被理解并且在开放源码软件包中广泛可用。许多其他复合体是其他物理领域的核心；例如，弹性复合体支撑固体力学，而海森复合体支撑爱因斯坦广义相对论方程。然而，人们对de Rham复合体以外的复合体的离散化知之甚少。只有有限元的部分知识是可用的，所得到的有限元是陌生的，在任何健壮的数学软件中都是不可用的。在这个项目中，我们将开发数学和计算工具来离散De Rham复形以外的复形。这项工作将在数值分析和计算机科学之间进行。该项目的第一步将是对目前已知的弹性和海森复合体的离散化进行彻底的审查。利用这些知识，我们将开发一种策略，使新的有限元能够快速实现。我们预计，这将涉及到在Firedrake软件中设计一种特定于领域的语言，以允许在代码中使用与数学规范同构的有限元软件规范。我们将验证我们开发的特定于领域的语言，使用它来实现已经手动实现的奇异元素(如Argyris和Arnold-Winther元素)，然后再讨论尚未提供的元素(如Zhang、Bogner-Fox-Schmidt和Hu-Zhang-Zhang元素)。一旦核心语言实现，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结构从德罗姆复形的几个副本导出(连续)弹性复形、黑森复形和其他复形。就像BGG结构可以在连续水平上应用一样，我们推测它也可以在离散水平上应用，从我们对如何离散De Rham复合体的知识中揭示了如何离散化其他复合体的洞察力。这项研究将使目前难以处理或需要具有严重缺陷的非协调离散的广泛的偏微分方程组的数值解成为可能，包括电磁学中的高阶旋度问题，近晶A液晶的Pevnyi-Selinger-SLuckin模型，以及超弹性固体材料的应力-位移公式。本项目属于EPSRC数值分析研究领域。

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.