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

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.