The de Rham complex on three-dimensional domains is the sequence of spaces H1, H(curl), H(div), and L2 connected by the vector calculus operators gradient, curl, divergence. The well-posedness of an important class of partial differential equations (PDEs) is closely linked to the cohomological properties of this complex, which characterize the kernel of each operator in terms of the topology of the domain. Reproducing these properties at the discrete level is crucial to obtain stable discretizations of such PDEs. A classical approach relies on trimmed (Raviart–Thomas–Nédélec) finite elements [1,2]. However, these finite elements are restricted to conforming meshes with elements of simple shape.
The efficiency of polyhedral meshes for the simulation of PDE models has been demonstrated in a number of settings, in particular for handling mesh refinement and coarsening. The Discrete De Rham (DDR) method [3] is an arbitrary-order discretization of the de Rham complex constructed on general polyhedral meshes. A crucial step in this construction consists in introducing a fully discrete formulation, in which both the spaces and the operators of the de Rham complex are replaced by discrete counterparts. I will show how such a formulation can be derived starting from the trimmed finite element complex and removing the need for globally defined function spaces.
Originally developed in the language of vector calculus, the DDR framework has recently been extended to the de Rham complex of differential forms [4]. This extension has enabled the derivation of general proofs of key properties, such as Poincaré inequalities and adjoint consistency [5–7].
[1] Raviart, P. A. and Thomas, J. M. (1977). A mixed finite element method for 2nd order elliptic problems. In Galligani, I. and Magenes, E., editors, Mathematical Aspects of the Finite Element Method. Springer, New York.
[2] Nédélec, J.-C. (1980). Mixed finite elements in R3. Numer. Math., 35(3):315–341.
[3] D. A. Di Pietro and J. Droniou. An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. Found. Comput. Math., 2023, 23:85–164. DOI: 10.1007/s10208-021-09542-8
[4] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods. J. Eur. Math. Soc., 2025. Published online. DOI: 10.4171/JEMS/1602
[5] D. A. Di Pietro and M.-L. Hanot. Uniform Poincaré inequalities for the Discrete de Rham complex on general domains. Results Appl. Math., 2024, 23(100496). DOI: 10.1016/j.rinam.2024.100496
[6] D. A. Di Pietro, J. Droniou, M.-L. Hanot, and S. Pitassi. Uniform Poincaré inequalities for the discrete de Rham complex of differential forms. arXiv preprint 2501.16116 [math.NA], January 2025
[7] D. A. Di Pietro, J. Droniou, and S. Pitassi. Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms. arXiv preprint 2509.21449 [math.NA], September 2025