  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.
