Finite element form-valued forms

Kaibo Hu (University of Edinburgh)

Abstract:
Classical finite element methods, such as those by Lagrange, Nédélec,
Raviart–Thomas, and Brezzi-Douglas-Marini, fit within de Rham complexes
and can be interpreted as discrete differential forms. These finite
element differential forms encode discrete topology and have become
standard practice for solving vector-valued problems. Their structures
also find broad applications in discrete topology, including
topological data analysis and the Hodge Laplacian on graphs.

In this work, we focus on tensors with applications in continuum
mechanics, differential geometry, and general relativity. First, we
investigate the algebraic and differential structures of tensor fields.
We show that tensor fields with natural symmetries fit within
Bernstein-Gelfand-Gelfand (BGG) complexes and twisted de Rham
complexes, and we discuss the correspondence between these complexes,
generalized continua, and Riemann-Cartan geometry. Second, we construct
finite elements for form-valued forms (double forms). Special cases
include classical finite element differential forms, distributional
finite elements, Christiansen’s finite element interpretation for Regge
calculus in quantum and numerical gravity (discrete metric and
curvature), the TDNNS/HHJ element for elasticity, the MCS element for
Stokes equations, and various new spaces.

References:

[1] Arnold, D. N., & Hu, K. (2021). Complexes from complexes.
Foundations of Computational Mathematics, 21(6), 1739-1774.

[2] Čap, A., & Hu, K. (2024). BGG sequences with weak regularity and
applications. Foundations of Computational Mathematics, 24(4), 1145-
1184.

[3] Hu, K., Lin, T., & Zhang, Q. (2025). Distributional Hessian and
divdiv complexes on triangulation and cohomology. SIAM Journal on
Applied Algebra and Geometry, 9(1), 108-153.

[4] Hu, K., & Lin, T. (2025). Finite element form-valued forms (I):
Construction. arXiv preprint arXiv:2503.03243.