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On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

Andrea Brugnoli, Volker Mehrmann

TL;DR

This work investigates whether discrete Lagrangian, Hamiltonian, and mixed finite element formulations for linear wave phenomena can be made equivalent at the fully discrete level. By combining conforming spatial discretizations with structure-preserving time integrators from the Newmark family (leapfrog and implicit midpoint), the authors show that, under compatibility conditions between FE spaces and with a trapezoidal reconstruction of the primal variable, the Lagrangian and mixed formulations, as well as their Hamiltonian interpretation, yield equivalent fully discrete schemes for both the wave equation and Maxwell equations. Key contributions include precise space-compatibility criteria, detailed semi-discrete equivalences for wave and Maxwell problems, and explicit mappings between time-discrete variants (velocity-stress vs. curl-based formulations) under leapfrog and implicit midpoint time stepping. The results have practical impact for designing robust, structure-preserving solvers for wave propagation, and provide a unified framework to analyze and compare FE formulations across Lagrangian, Hamiltonian, and port-Hamiltonian perspectives.

Abstract

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

TL;DR

This work investigates whether discrete Lagrangian, Hamiltonian, and mixed finite element formulations for linear wave phenomena can be made equivalent at the fully discrete level. By combining conforming spatial discretizations with structure-preserving time integrators from the Newmark family (leapfrog and implicit midpoint), the authors show that, under compatibility conditions between FE spaces and with a trapezoidal reconstruction of the primal variable, the Lagrangian and mixed formulations, as well as their Hamiltonian interpretation, yield equivalent fully discrete schemes for both the wave equation and Maxwell equations. Key contributions include precise space-compatibility criteria, detailed semi-discrete equivalences for wave and Maxwell problems, and explicit mappings between time-discrete variants (velocity-stress vs. curl-based formulations) under leapfrog and implicit midpoint time stepping. The results have practical impact for designing robust, structure-preserving solvers for wave propagation, and provide a unified framework to analyze and compare FE formulations across Lagrangian, Hamiltonian, and port-Hamiltonian perspectives.

Abstract

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.
Paper Structure (11 sections, 2 theorems, 47 equations)

This paper contains 11 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Continuous formulations
  3. Wave equation
  4. Maxwell equations
  5. Semi-discretization in space
  6. The semi-discrete wave equation
  7. The semi-discrete Maxwell equations
  8. Time discretization
  9. The leapfrog scheme
  10. The implicit midpoint rule
  11. Conclusion

Key Result

Proposition 1

Suppose that $\sigma_{h, M}^0 = k \nabla q_{h, L}^0$ and that the field $q_{h, M}^{n+1}$ with $q_{h, M}^{0}= q_{h, L}^{0}$ is reconstructed via the trapezoidal rule then the formulations eq:wave_first_sv and eq:wave_mixed_grad_sv are equivalent.

Theorems & Definitions (4)

  • Proposition 1
  • Proof 1
  • Proposition 2
  • Proof 2