  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. In this work, we develop an
additive multigrid preconditioner for the fractional Laplacian with positive
fractionality, and show a uniform bound on the condition number. For the case
of negative fractionality, we re-use the preconditioner developed for the
positive fractionality and left-right multiply a regular Laplacian with a
preconditioner with positive fractionality to obtain the desired negative
fractionality. Implementational issues are outlined in details as the
differences between the discrete operators and their corresponding matrices
must be addressed when realizing these algorithms in code. We finish with some
numerical experiments verifying the theoretical findings.
