Finite element hybridization of port-Hamiltonian systems
Authors
Andrea Brugnoli, Ramy Rashad, Yi Zhang, Stefano Stramigioli
Abstract
In this contribution, we extend the hybridization framework for the Hodge Laplacian [Awanou et al. (2023) [16]] 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. The mixed formulation is then hybridized to obtain an equivalent formulation that can be more 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 the final system only contains the globally coupled traces of one variable. Numerical experiments on the 3D wave and Maxwell equations illustrate the convergence of the method and the size reduction achieved by the hybridization.
Keywords
Port-Hamiltonian systems; Finite element exterior calculus; Hybridization; Dual field
Citation
- Journal: Applied Mathematics and Computation
- Year: 2025
- Volume: 498
- Issue:
- Pages: 129377
- Publisher: Elsevier BV
- DOI: 10.1016/j.amc.2025.129377
BibTeX
@article{Brugnoli_2025,
title={{Finite element hybridization of port-Hamiltonian systems}},
volume={498},
ISSN={0096-3003},
DOI={10.1016/j.amc.2025.129377},
journal={Applied Mathematics and Computation},
publisher={Elsevier BV},
author={Brugnoli, Andrea and Rashad, Ramy and Zhang, Yi and Stramigioli, Stefano},
year={2025},
pages={129377}
}
References
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- Rashad, Twenty years of distributed port-Hamiltonian systems: a literature review. IMA J. Math. Control Inf. (072020)
- Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
- Hiptmair, R. Discrete Hodge operators. Numerische Mathematik vol. 90 265–289 (2001) – 10.1007/s002110100295
- Hirani, (2003)
- Kumar,
- Kapidani, B. & Hernandez, R. V. High Order Geometric Methods With Splines: An Analysis of Discrete Hodge-Star Operators. SIAM Journal on Scientific Computing vol. 44 A3673–A3699 (2022) – 10.1137/22m1481762
- Brugnoli, A., Rashad, R. & Stramigioli, S. Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics vol. 471 111601 (2022) – 10.1016/j.jcp.2022.111601
- Arnold, D. N., Falk, R. S. & Winther, R. Finite element exterior calculus, homological techniques, and applications. Acta Numerica vol. 15 1–155 (2006) – 10.1017/s0962492906210018
- Wu, Y. & Bai, Y. Error Analysis of Energy-Preserving Mixed Finite Element Methods for the Hodge Wave Equation. SIAM Journal on Numerical Analysis vol. 59 1433–1454 (2021) – 10.1137/19m1307950
- Patrick, (2003)
- Cockburn, B., Gopalakrishnan, J. & Lazarov, R. Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems. SIAM Journal on Numerical Analysis vol. 47 1319–1365 (2009) – 10.1137/070706616
- GUYAN, R. J. Reduction of stiffness and mass matrices. AIAA Journal vol. 3 380–380 (1965) – 10.2514/3.2874
- Park, K. C. et al. Displacement‐based partitioned equations of motion for structures: Formulation and proof‐of‐concept applications. International Journal for Numerical Methods in Engineering vol. 124 5020–5046 (2023) – 10.1002/nme.7334
- González, J. A. & Park, K. C. Three-field partitioned analysis of fluid–structure interaction problems with a consistent interface model. Computer Methods in Applied Mechanics and Engineering vol. 414 116134 (2023) – 10.1016/j.cma.2023.116134
- Awanou, G., Fabien, M., Guzmán, J. & Stern, A. Hybridization and postprocessing in finite element exterior calculus. Mathematics of Computation vol. 92 79–115 (2022) – 10.1090/mcom/3743
- Cohen, (2002)
- Egger, H. & Radu, B. A Second-Order Finite Element Method with Mass Lumping for Maxwell’s Equations on Tetrahedra. SIAM Journal on Numerical Analysis vol. 59 864–885 (2021) – 10.1137/20m1318912
- Kirby, R. C. & Kieu, T. T. Symplectic-mixed finite element approximation of linear acoustic wave equations. Numerische Mathematik vol. 130 257–291 (2014) – 10.1007/s00211-014-0667-4
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
- Kreeft,
- Frankel, (2011)
- Weck, N. TRACES OF DIFFERENTIAL FORMS ON LIPSCHITZ BOUNDARIES. Analysis vol. 24 (2004) – 10.1524/anly.2004.24.14.147
- Schulz, (2022)
- Arnold, (2018)
- Weck, N. TRACES OF DIFFERENTIAL FORMS ON LIPSCHITZ BOUNDARIES. Analysis vol. 24 (2004) – 10.1524/anly.2004.24.14.147
- Bacuta, C., Demkowicz, L., Mora, J. & Xenophontos, C. Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms. Computers & Mathematics with Applications vol. 95 215–241 (2021) – 10.1016/j.camwa.2020.09.018
- Wu, Y. & Bai, Y. Error Analysis of Energy-Preserving Mixed Finite Element Methods for the Hodge Wave Equation. SIAM Journal on Numerical Analysis vol. 59 1433–1454 (2021) – 10.1137/19m1307950
- Palha, A., Rebelo, P. P., Hiemstra, R., Kreeft, J. & Gerritsma, M. Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. Journal of Computational Physics vol. 257 1394–1422 (2014) – 10.1016/j.jcp.2013.08.005
- Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
- Sanz-Serna, J. M. Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica vol. 1 243–286 (1992) – 10.1017/s0962492900002282
- Güdücü, C., Liesen, J., Mehrmann, V. & Szyld, D. B. On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs. SIAM Journal on Scientific Computing vol. 44 A2871–A2894 (2022) – 10.1137/21m1458594
- Rathgeber, F. et al. Firedrake. ACM Transactions on Mathematical Software vol. 43 1–27 (2016) – 10.1145/2998441
- Gibson, T. H., Mitchell, L., Ham, D. A. & Cotter, C. J. Slate: extending Firedrake’s domain-specific abstraction to hybridized solvers for geoscience and beyond. Geoscientific Model Development vol. 13 735–761 (2020) – 10.5194/gmd-13-735-2020
- Douglas, J. & Roberts, J. E. Global estimates for mixed methods for second order elliptic equations. Mathematics of Computation vol. 44 39–52 (1985) – 10.1090/s0025-5718-1985-0771029-9
- Grote, M. J., Schneebeli, A. & Schötzau, D. Discontinuous Galerkin Finite Element Method for the Wave Equation. SIAM Journal on Numerical Analysis vol. 44 2408–2431 (2006) – 10.1137/05063194x
- Jain, V., Zhang, Y., Palha, A. & Gerritsma, M. Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes. Computers & Mathematics with Applications vol. 95 101–142 (2021) – 10.1016/j.camwa.2020.09.022
- Rashad, R., Brugnoli, A., Califano, F., Luesink, E. & Stramigioli, S. Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation. Journal of Nonlinear Science vol. 33 (2023) – 10.1007/s00332-023-09945-7
- Rashad, R. & Stramigioli, S. The Port-Hamiltonian Structure of Continuum Mechanics. Journal of Nonlinear Science vol. 35 (2025) – 10.1007/s00332-025-10130-1