We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Br\'ezis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Br\'ezis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lam\'e equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.
H. Antil, Alex Kaltenbach, Keegan L. A. Kirk· 0 citations
Abstract We study Maxwell’s equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and L 2-data. Our first contribution is a direct proof of well-posedness (existence, uniqueness and continuous dependence on data) of the first-order weak formulation. An energy identity is also derived. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf. Duvaut and Lions [G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, volume 219 of Grundlehren der Mathematischen Wissenschaften, Berlin-New York, Springer-Verlag, 1976. Translated from the French by C. W. John, Eqns. (4.31)–(4.32), p. 346; Thm. 4.1]). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the Nédélec/Raviart–Thomas de Rham complex. The semi-discrete problem is shown to be well-posed. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained L 2 projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded L 2 spaces, and identification of time derivatives in dual spaces. The results at the semi-discrete level are novel.
H. Antil· Journal of Numerical Mathema...· 0 citations
We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.
H. Antil, Rocío Díaz Martín, Ivan V. Medri et al.· arXiv.org· 0 citations
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