  We present a monolithic geometric multigrid preconditioner for solving
fluid-solid interaction problems in Stokes limit. The problems are discretized
by a spatially adaptive high-order meshless method, the generalized moving
least squares (GMLS) with adaptive $h$-refinement. In Stokes limit, solid
kinematics can be dominated by the singularities governing the lubrication
effects. Resolving those singularities with adaptive $h$-refinement can lead to
an ill-conditioned linear system of equations. For constructing the
interpolation and restriction operators - the key ingredients of the multigrid
preconditioner, we utilize the geometric information of hierarchical sets of
GMLS nodes generated in adaptive $h$-refinement. We build decoupled smoothers
through physics-based splitting and then combine them via a multiplicative
overlapping Schwarz approach. Through numerical examples with the inclusion of
different numbers and shapes of solid bodies, we demonstrate the performance
and assess the scalability of the designed preconditioner. As the total degrees
of freedom and the number of solid bodies $N_s$ increase, the proposed
monolithic geometric multigrid preconditioner can ensure convergence and good
scalability when using the Krylov iterative method for solving the linear
systems of equations generated from the spatially adaptive GMLS discretization.
More specifically, for a fixed number of solid bodies, as the discretization
resolution is incrementally refined, the number of iterations of the linear
solver can be maintained at the same level, indicating nearly linear
scalability of our preconditioner with respect to the total degrees of freedom.
When $N_s$ increases, the number of iterations is nearly proportional to
$\sqrt{N_s}$, implying the sublinear optimality with respect to the number of
solid bodies.
