Kaibo Hu - Finite element form-valued forms

This lecture was part of the Thematic Programme on "Differential Complexes: Theory, Discretization, and Applications" held at the ESI April 20 — June 5, 2026.

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.

These constructions aim to address the problem of discretizing tensor fields with general symmetries on a triangulation, with a particular focus on establishing discrete differential-geometric structures and compatible tensor decompositions in 2D, 3D, and higher dimensions.

This is a joint work with Ting Lin (Peking University).

Видео Kaibo Hu - Finite element form-valued forms канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)