Finite element hybridization of port-Hamiltonian systems

TL;DR

This paper extends the hybridization framework to port-Hamiltonian systems for wave phenomena, introducing a dual field mixed Galerkin discretization that preserves physical laws and enables efficient solution via static condensation.

Contribution

It develops a novel hybridization approach for port-Hamiltonian systems, combining conforming and local finite element spaces, with proven conservation properties and improved computational efficiency.

Findings

The method converges for 3D wave and Maxwell equations.

Hybridization achieves significant size reduction.

The scheme preserves discrete power balance and conservation laws.

Abstract

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 describing linear wave propagation phenomena. To this aim, a dual field mixed Galerkin discretization is introduced, in which one variable is approximated via conforming finite element spaces, whereas the second is completely local. This scheme is equivalent to the second order mixed Galerkin formulation and retains a discrete power balance and discrete conservation laws. The mixed formulation is also equivalent to the hybrid formulation. 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 convergence of the method and the size reduction achieved by the hybridization.