Finite elements beyond the de Rham complex

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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.