  Coupled multiphysics problems often give rise to interface conditions
naturally formulated in fractional Sobolev spaces. Here, both positive and
negative fractionality are common. When designing efficient solvers for
discretizations of such problems it would then be useful to have a
preconditioner for the fractional Laplacian, $(-\Delta)^s$, with $s \in
[-1,1]$. Previously, additive multigrid preconditioners for the case when $s
\geq 0$ have been proposed. In this work we complement this construction with
auxiliary space preconditioners suitable when $s \leq 0$. These preconditioners
are shown to be spectrally equivalent to $(-\Delta)^{-s}$, but requires
preconditioners for fractional $H(\operatorname{div})$ operators with positive
fractionality. We design such operators based on an additive multigrid
approach. We finish with some numerical experiments, verifying the theoretical
results.
