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arXiv · 2607.13256

Duality Framework for Flux Constrained Flow in Porous Media: Analysis and Numerics

Harbir Antil·Keegan L. A. Kirk·Felipe Pérez
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Abstract

We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};\Omega)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal--dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart--Thomas/Crouzeix--Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal--dual gap indicators, support the theory.

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BibTeXRIS

Harbir Antil, Keegan L. A. Kirk, Felipe Pérez. 2026-07-14. Duality Framework for Flux Constrained Flow in Porous Media: Analysis and Numerics. https://arxiv.org/abs/2607.13256

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