Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

• Published in: arXiv.org
• Main Authors: Kumar, N, J J W van der Vegt, Zwart, H J
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 14.12.2022
• Subjects: Computer Science - Numerical Analysis; Conservation laws; Dynamical systems; Energy flow; Error analysis; Finite element analysis; Finite element method; Function space; Galerkin method; Mathematics - Analysis of PDEs; Mathematics - Dynamical Systems; Mathematics - Numerical Analysis; Wave equations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2212.07041

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.