Extended finite elements for 3D–1D coupled problems via a PDE-constrained optimization approach

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.