Finite element hybridization of port-Hamiltonian systems

In this contribution, we extend the hybridization framework for the Hodge Laplacian (Awanou et al., Hybridization and postprocessing in finite element exterior calculus, 2023) to port-Hamiltonian systems. To this aim, a general dual field continuous Galerkin discretization is introduced, in which one variable is approximated via conforming finite element spaces, whereas the second is completely local. This scheme retains a discrete power balance and discrete conservation laws and is directly amenable to hybridization. The hybrid formulation is equivalent to the continuous Galerkin formulation and to a power preserving interconnection of port-Hamiltonian systems, thus providing a system theoretic interpretation of finite element assembly. The hybrid system can be efficiently solved using a static condensation procedure in discrete time. The size reduction achieved thanks to the hybridization is greater than the one obtained for the Hodge Laplacian as one field is completely discarded. Numerical experiments on the 3D wave and Maxwell equations show the equivalence of the continuous and hybrid formulation and the computational gain achieved by the latter.