AI 中文总结
本文基于Brézis-Ekeland-Nayroles原理推导次梯度流的基于对偶性的后验误差恒等式，将其应用于多个物理相关非稳态变分问题，为次梯度流的误差分析提供了新框架。
AI 中文摘要
我们针对由依赖时间的凸积分泛函诱导的一大类次梯度流，推导了基于对偶性的后验误差恒等式。从Brézis-Ekeland-Nayroles原理出发，我们确定了一个非稳态原始能量泛函，并推导了其Fenchel对偶形式，包括在一般法向积分项假设下的强对偶性及对应的最优性系统。该Fenchel对偶框架被用于推导次梯度流的后验误差恒等式。在此过程中，我们脱离了非稳态情形下常规的基于对偶性的后验误差控制框架，因为Brézis-Ekeland-Nayroles公式揭示了如下非稳态特征：最小原始值和最大对偶值均由初始数据确定。这使我们能够从组合的原始-对偶间隙恒等式过渡到独立的原始和对偶间隙恒等式。这些恒等式独立量化原始和对偶误差，并以广义Bregman散度的形式呈现，且在空间凸共轭公式下，呈现为适合局域化的非负时空积分量。该抽象框架被应用于多个具有物理意义的变分问题，包括非稳态热方程、非稳态Stokes方程、非稳态Navier-Lamé方程、非稳态管道Bingham流、非稳态障碍问题以及非稳态弹塑性扭转问题。
英文摘要
We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Brézis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Brézis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lamé equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.
Comments46 pages
