  Ill-conditioning of the system matrix is a well-known complication in
immersed finite element methods and trimmed isogeometric analysis. Elements
with small intersections with the physical domain yield problematic eigenvalues
in the system matrix, which generally degrades efficiency and robustness of
iterative solvers. In this contribution we investigate the spectral properties
of immersed finite element systems treated by Schwarz-type methods, to
establish the suitability of these as smoothers in a multigrid method. Based on
this investigation we develop a geometric multigrid preconditioner for immersed
finite element methods, which provides mesh-independent and
cut-element-independent convergence rates. This preconditioning technique is
applicable to higher-order discretizations, and enables solving large-scale
immersed systems in parallel, at a computational cost that scales linearly with
the number of degrees of freedom. The performance of the preconditioner is
demonstrated for conventional Lagrange basis functions and for isogeometric
discretizations with both uniform B-splines and locally refined approximations
based on truncated hierarchical B-splines.
