Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

中文翻译：

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分层混合维椭圆方程的后验误差估计

具有层次结构的混合维椭圆方程通常用于模拟具有高纵横比夹杂物的问题，例如破裂多孔介质中的流动。我们基于泛函理论推导出一般抽象估计后验的误差估计，获得了原始变量和对偶变量的保证上限以及原始对偶对的两侧边界。我们通过根据近似解具有守恒性质的程度提出四种不同的估计残差误差的方法来改进用函数方法获得的抽象结果，即：（1）不守恒，（2）子域守恒，（3）网格级守恒，和 (4) 精确守恒。当质量在网格级别或完全守恒时，这种处理会产生更清晰和完全可计算的估计，并且具有与从基于网格获得的结构相当的结构后验的技巧。我们通过数值实验证明了我们的理论结果的实际有效性，使用四种不同的离散化方法来解决基于裂缝多孔介质流动基准的合成问题和应用。