Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

The paper’s Theorem 8.3.1 proves the a priori L2-error estimate for the port-Hamiltonian DG scheme (136), using smoothed FEEC projections that commute with d, an energy identity, face-term cancellations via the conservative θ-flux, and Grönwall—yielding inequality (159) with the claimed O(h^{2(r+1)}) rate under standard regularity. The candidate solution follows the same architecture: projection-based error splitting, commuting-projection cancellations of volume terms, control of skeleton residuals with trace inequalities, a weighted/discrete Hodge–star energy, Young + Grönwall, and insertion of FEEC approximation bounds. This aligns closely with the paper’s Lemmas 8.1.1–8.1.2 and the energy equation setup (156)–(158), culminating in (175)–(176) and then (159) . Minor differences are present in notation and the explicit mention of spectral equivalence of discrete Hodge stars (implicit in the paper’s weighted inner products and coefficients Cp, Cq) and the use of generic DG trace/ inverse-trace estimates; these are standard and consistent with the paper’s framework.