  Coupled partial differential equations defined on domains with different
dimensionality are usually called mixed dimensional PDEs. We address mixed
dimensional PDEs on three-dimensional (3D) and one-dimensional domains, giving
rise to a 3D-1D coupled problem. Such problem poses several challenges from the
standpoint of existence of solutions and numerical approximation. For the
coupling conditions across dimensions, we consider the combination of essential
and natural conditions, basically the combination of Dirichlet and Neumann
conditions. To ensure a meaningful formulation of such conditions, we use the
Lagrange multiplier method, suitably adapted to the mixed dimensional case. The
well posedness of the resulting saddle point problem is analyzed. Then, we
address the numerical approximation of the problem in the framework of the
finite element method. The discretization of the Lagrange multiplier space is
the main challenge. Several options are proposed, analyzed and compared, with
the purpose to determine a good balance between the mathematical properties of
the discrete problem and flexibility of implementation of the numerical scheme.
The results are supported by evidence based on numerical experiments.
