| Issue |
ESAIM: M2AN
Volume 59, Number 3, May-June 2025
|
|
|---|---|---|
| Page(s) | 1565 - 1600 | |
| DOI | https://doi.org/10.1051/m2an/2025035 | |
| Published online | 04 June 2025 | |
Finite elements for Wasserstein 𝕎p gradient flows
1
Univ. Lille, CNRS, Inria, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
2
Zentrum für Mathematik, Technische Universit¨at München, 85747 Garching, Germany
3
CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
* Corresponding author: clement.cances@inria.fr
Received:
30
April
2024
Accepted:
1
May
2025
Wasserstein 𝕎p gradient flows for nonlinear integral functionals of the density yield degenerate parabolic equations involving diffusion operators of q-Laplacian type, with q being p’s conjugate exponent. We propose a finite element scheme building on conformal 𝕡1 Lagrange elements with mass lumping and a backward Euler time discretization strategy. Our scheme preserves mass and positivity while energy decays in time. Building on the theory of gradient flows in metric spaces, we further prove convergence towards a weak solution of the PDE that satisfies the energy dissipation equality. The analytical results are illustrated by numerical simulations.
Mathematics Subject Classification: 65M12 / 65M60 / 35K65 / 35K92
Key words: Wasserstein gradient flow / finite elements / nonlinear stability / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2025
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