AI 中文总结
研究多孔介质中通量约束流动问题，通过芬切尔对偶性建立基于速度的双凹能量最大化问题与基于压力的预对偶公式，推导误差恒等式和先验误差衰减率，数值实验验证理论，为该领域提供对偶框架及分析方法。
AI 中文摘要
我们引入并分析了通过饱和多孔介质的达西流，该介质在边界的一部分上受到法向通量的双边约束。该问题被设定为基于速度的双凹能量在\(H(\mathrm{div};\Omega)\)的凸子集上的最大化；芬切尔对偶性确定了基于压力的预对偶公式，产生强对偶性，并在约束边界上提供具有互补结构的凸最优性条件。原始 - 对偶间隙满足一个后验误差恒等式，不含通用常数，对任意可允许的近似都有效。对偶结构由拉维亚特 - 托马斯/克鲁泽克斯 - 拉维亚特离散化继承，我们从中推导出离散误差恒等式和在解和通量边界的分数正则性假设下的先验误差衰减率。数值实验，包括由局部原始 - 对偶间隙指标驱动的自适应细化，支持了该理论。
英文摘要
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.
