Port-Hamiltonian discontinuous Galerkin finite element methods

Authors

Nishant Kumar, J J W van der Vegt, H J Zwart

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 nonzero 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 semidiscrete 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.

Citation

• Journal: IMA Journal of Numerical Analysis
• Year: 2025
• Volume: 45
• Issue: 1
• Pages: 354–403
• Publisher: Oxford University Press (OUP)
• DOI: 10.1093/imanum/drae008

BibTeX

@article{Kumar_2024,
  title={{Port-Hamiltonian discontinuous Galerkin finite element methods}},
  volume={45},
  ISSN={1464-3642},
  DOI={10.1093/imanum/drae008},
  number={1},
  journal={IMA Journal of Numerical Analysis},
  publisher={Oxford University Press (OUP)},
  author={Kumar, Nishant and van der Vegt, J J W and Zwart, H J},
  year={2024},
  pages={354--403}
}

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