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Coupled multiphysics problems often give rise to interface conditionsnaturally formulated in fractional Sobolev spaces. Here, both positive andnegative fractionality are common. When designing efficient solvers fordiscretizations of such problems it would then be useful to have apreconditioner 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 withauxiliary space preconditioners suitable when $s \leq 0$. These preconditionersare shown to be spectrally equivalent to $(-\Delta)^{-s}$, but requirespreconditioners for fractional $H(\operatorname{div})$ operators with positivefractionality. We design such operators based on an additive multigridapproach. We finish with some numerical experiments, verifying the theoreticalresults.

An Auxiliary Space Preconditioner for Fractional Laplacian of Negative Order