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，例如裂缝性多孔介质中的流动、纤维增强材料的研究或生物组织微循环建模（仅举几例），可描述为定义在相互嵌入的异质维度域中的耦合偏微分方程。这一表述源于几何模型降阶技术，该技术将原始定义在复杂三维域中的问题转化为更易处理的形式。适用于此类问题的耦合算子的定义与近似仍是一个挑战。我们建立了一个通用数学框架，用于分析及近似不同维度间通过非匹配约束耦合的偏微分方程，重点关注其基于拉格朗日乘子的实现。在此背景下，我们从抽象且通用的角度探讨了问题相对于嵌入域最小特征长度的适定性、稳定性及鲁棒性。同时，我们讨论了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。本研究的主旨包含两方面：从混合维度问题理论角度，我们提供了用于构建跨维度耦合问题的通用抽象数学工具；从数值近似实践角度，我们展示了在若干对应用具重要性的代表性构型中，网格特征尺寸、拉格朗日乘子空间维度与嵌入体尺寸之间的相互作用。最后，通过数值算例补充了对后者的分析。