AI 中文总结
本文提出了模拟三维扩散固体内嵌一维扩散纤维耦合行为的混合维度扩散问题模型，给出有限元近似并证明其收敛性，数值例子验证了理论结果。
AI 中文摘要
我们提出了一个边值问题的公式，用于模拟嵌入三维扩散固体中的一维扩散纤维的耦合行为。我们引入该问题的变分表述，将关联的扩散场识别为耦合约束下的能量极小化子，证明该鞍点问题适定。随后，我们对所提出的边值问题引入有限元离散化，证明有限元解收敛到精确解。该近似最显著的特征是各物体的网格无需协调，数值例子验证了理论结果。
英文摘要
We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results
