In this work, we propose the application of the eXtended Finite Element Method (XFEM) in the context of the coupling between three-dimensional and one-dimensional elliptic problems. In particular, we consider the case in which the 3D-1D coupled problem arises from the geometrical model reduction of a fully three-dimensional problem, characterized by thin tubular inclusions embedded in a much wider domain. In the 3D-1D coupling framework, the use of non conforming meshes is widely adopted. However, since the inclusions typically behave as singular sinks or sources for the 3D problem, mesh adaptation near the embedded 1D domains may be necessary to enhance solution accuracy and recover optimal convergence rates. An alternative to mesh adaptation is represented by the XFEM, which we here propose to enhance the approximation capabilities of an optimization-based 3D-1D coupling approach. An effective quadrature strategy is devised to integrate the enrichment functions and numerical tests on single and multiple segments are proposed to demonstrate the effectiveness of the approach.

翻译：本文提出将扩展有限元方法（XFEM）应用于三维与一维椭圆问题的耦合场景。具体而言，我们考虑由全三维问题的几何模型降阶而产生的3D-1D耦合问题，其特征为薄管状夹杂物嵌入于更宽广的域中。在3D-1D耦合框架下，非协调网格的广泛使用已成为常态。然而，由于夹杂物通常表现为三维问题的奇异汇点或源点，为提升解精度并恢复最优收敛率，需在嵌入的一维域附近进行网格自适应。作为网格自适应的替代方案，本文提出采用XFEM增强基于优化的3D-1D耦合方法的逼近能力。我们设计了有效的求积策略用于积分富集函数，并通过单段与多段数值算例验证了该方法的有效性。