  We propose nodal auxiliary space preconditioners for facet and edge virtual
elements of lowest order by deriving discrete regular decompositions on
polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual
element framework. The preconditioner consists of solving a sequence of
elliptic problems on the nodal virtual element space, combined with appropriate
smoother steps. Under assumed regularity of the mesh, the preconditioned system
is proven to have bounded spectral condition number independent of the mesh
size and this is verified by numerical experiments on a sequence of polygonal
meshes. Moreover, we observe numerically that the preconditioner is robust on
meshes containing elements with high aspect ratios.
