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.

翻译：本文我们将Hodge Laplacian的混合化框架（Awanou等人，2023年，《有限元外微分中的混合化与后处理》）扩展至port-Hamiltonian系统。为此，引入了一种通用的双场连续Galerkin离散格式，其中第一个变量通过协调有限元空间近似，而第二个变量完全局部化。该格式保留了离散功率平衡与离散守恒律，并可直接进行混合化。混合化公式等价于连续Galerkin公式，也等价于port-Hamiltonian系统的功率保持互联，从而为有限元组装提供了系统理论解释。该混合系统可通过离散时间下的静态凝聚过程高效求解。由于其中一个场被完全消除，相比于Hodge Laplacian的混合化，本方法实现了更大的规模缩减。针对三维波动方程和Maxwell方程的数值实验验证了连续公式与混合化公式的等价性，并展示了后者带来的计算增益。