Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues -- just to mention a few examples -- can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the inf-sup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.

翻译：许多涉及异质空间尺度的物理问题——例如裂缝多孔介质中的流动、纤维增强材料的研究、或活体组织微循环的模拟（仅举几例）——可描述为定义在相互嵌入的异质维度域上的耦合偏微分方程。这种表述源于几何模型约化技术，该技术将原定义于复杂三维空间的问题转化为更易处理的形式。针对该类问题定义并近似合适的耦合算子仍是挑战。我们建立了一个通用数学框架，用于分析和近似受非匹配约束跨维度耦合的偏微分方程，重点关注其基于拉格朗日乘子的强制实现。在此背景下，我们以抽象且通用的方式讨论了问题关于嵌入域最小特征长度的适定性、稳定性与鲁棒性。还讨论了该问题的数值近似方法，并针对嵌入域的若干代表性构型，论证了所提数值格式的inf-sup稳定性。本研究主要传递双重信息：从混合维度问题理论角度，我们提供了跨维度耦合问题公式化的通用抽象数学工具；从数值近似的实践角度，我们揭示了在若干具应用代表性的构型中，网格特征尺寸、拉格朗日乘子空间维度与包含体尺寸之间的相互作用。后者辅以说明性数值算例加以验证。