  This work introduces nodal auxiliary space preconditioners for
discretizations of mixed-dimensional partial differential equations. We first
consider the continuous setting and generalize the regular decomposition to
this setting. With the use of conforming mixed finite element spaces, we then
expand these results to the discrete case and obtain a decomposition in terms
of nodal Lagrange elements. In turn, nodal preconditioners are proposed
analogous to the auxiliary space preconditioners of Hiptmair and Xu (2007).
Numerical experiments show the performance of this preconditioner in the
context of flow in fractured porous media.
