Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

TL;DR

This paper introduces a novel port-Hamiltonian discontinuous Galerkin finite element framework that preserves structure and energy flow, enabling accurate discretization of infinite-dimensional Hamiltonian systems with boundary interactions.

Contribution

It develops a new structure-preserving DG discretization for port-Hamiltonian systems, including a mathematical framework and error analysis, applicable to complex geometries.

Findings

Demonstrates optimal convergence rates for scalar wave equation

Provides a power-preserving coupling framework for DG discretizations

Establishes a priori error estimates for the proposed method

Abstract

A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a non-zero energy flow through the boundaries. In this paper we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes-Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.