Duality Framework for Flux Constrained Flow in Porous Media: Analysis and Numerics
Provides a rigorous duality-based analysis and numerical method for flux-constrained porous media flow, which is a specific problem in computational geoscience.
The paper introduces a duality framework for Darcy flow with bilateral flux constraints on the boundary, deriving strong duality, a posteriori error identities, and a priori error rates. Numerical experiments with adaptive refinement validate the theory.
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 $H(\mathrm{div};Ω)$; 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.