Finite element hybridization of port-Hamiltonian systems

TL;DR

This work extends the hybridization framework for Hodge-wave (Hodge Laplacian) problems to port-Hamiltonian systems modeling linear wave propagation. It adopts a dual-field mixed Galerkin discretization with one conforming and one local variable, proving equivalence to the second-order formulation and preserving a discrete power balance. The hybridization, coupled with static condensation and implicit midpoint time stepping, yields a substantial reduction in global degrees of freedom while maintaining the Dirac-structure-based energy conservation, as demonstrated on 3D wave and Maxwell problems. The results indicate strong convergence and significant computational savings, highlighting the method's practicality for modular port-Hamiltonian modeling and large-scale wave simulations.

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.

Figures (11)

• Figure 1: Illustration of global and local variables for $n=2, p=2, q=1$ for the lowest order polynomial mesh. The global variable lives on the mesh's skeleton and its degrees of freedom are in blue . The degrees of freedom of the local variable that undergoes hybridization is depicted in orange and those of the variable that is discontinuous and does not require regularity is depicted in purple .
• Figure 2: Equivalence of vector and exterior calculus Sobolev spaces.
• Figure 3: Equivalence between finite element differential forms and classical elements.
• Figure 4: Convergence rate for the different variables in the primal formulation of the wave equation, measured at $T_{\text{end}}=1$ for $\Delta t = \frac{1}{500}$.
• Figure 5: Convergence rate for the different variables in the dual formulation of the wave equation, measure at at $T_{\text{end}}=1$ for $\Delta t = \frac{1}{500}$.

Theorems & Definitions (14)

• Remark 1
• Proposition 1
• Remark 2
• Remark 3: Equivalence with the second order formulation
• Remark 4: Equivalence of the two conforming formulations
• Proposition 2
• proof
• Theorem 1
• proof
• Remark 5