  A numerical method for coupled 3D-1D problems with discontinuous solutions at
the interfaces is derived and discussed. This extends a previous work on the
subject where only continuous solutions were considered. Thanks to properly
defined function spaces a well posed 3D-1D problem is obtained from the
original fully 3D problem and the solution is then found by a PDE-constrained
optimization reformulation. This is a domain decomposition strategy in which
unknown interface variables are introduced and a suitably defined cost
functional, expressing the error in fulfilling interface conditions, is
minimized constrained by the constitutive equations on the subdomains. The
resulting discrete problem is robust with respect to geometrical complexity
thanks to the use of independent discretizations on the various subdomains.
Meshes of different sizes can be used without affecting the conditioning of the
discrete linear system, and this is a peculiar aspect of the considered
formulation. An efficient resolution strategy is further proposed, based on the
use of a gradient based solver and yielding a method ready for parallel
implementation. A numerical experiment on a problem with known analytical
solution shows the accuracy of the method, and two examples on more complex
configurations are proposed to address the applicability of the approach to
practical problems.
