Abstract:

Some of the most successful vector-valued finite elements in computational electromagnetics and fluid mechanics, such as the Nédélec and Raviart-Thomas elements, are recognized as special cases of Whitney’s discrete differential forms. Recent efforts aim to go beyond differential forms and establish canonical discretizations for more general tensors. An important class is that of form-valued forms, or double forms, which includes the metric tensor (symmetric (1,1)-forms) and the curvature tensor (symmetric (2,2)-forms). Like the differential structure of forms is encoded in the de Rham complex, that of double forms is encoded in the Bernstein–Gelfand–Gelfand (BGG) sequences and their cohomologies. Important examples include the Calabi complex in geometry and the Kröner complex in continuum mechanics.

In this talk, we first review the Nédélec and Raviart-Thomas elements and their generalizations to differential forms. We then discuss double forms, the corresponding BGG sequences, and their finite element versions. As an application, we reformulate the linearized Einstein equations (in the ADM formalism) as a Hodge–Dirac system. Within this framework, the well-posedness of the problem and the propagation of constraints are naturally encoded in both the continuous formulation and its discrete counterpart.

Bio:

Kaibo Hu is an Associate Professor and Senior Research Fellow at the University of Oxford, as well as a Royal Society University Research Fellow. His primary research interests include compatible and structure-preserving discretization, finite element exterior calculus, computational topological (magneto)hydrodynamics, and their applications.

He received his Bachelor's degree from Nankai University in 2012 and Ph.D. from Peking University in 2017. He conducted postdoctoral research at the University of Oslo and the University of Minnesota. From 2021 to 2023, he served as a Hooke Research Fellow at the University of Oxford, before being appointed as a Reader at the University of Edinburgh.

In 2023, he received the SIAM Early Career Prize in Computational Science and Engineering. In 2024, he was awarded an ERC Starting Grant “GeoFEM (Geometric Finite Element Methods)" from the European Research Council. He will deliver the Nachdiplom lectures at ETH Zurich in 2026. His publications have been selected for the SIAM High Impact Article Collection (2023) and received the Frontiers of Science Award from ICBS (2025), among several other paper prizes.