  This contribution presents a hierarchical multigrid approach for the solution
of large-scale finite cell problems on both uniform grids and multi-level
hp-discretizations. The proposed scheme leverages the hierarchical nature of
the basis functions utilized in the finite cell method and the multi-level
hp-method, which is attributed to the use of high-order integrated Legendre
basis functions and overlay meshes, to yield a simple and elegant multigrid
scheme. This simplicity is reflected in the fact that all restriction and
prolongation operators reduce to binary matrices that do not need to be
explicitly constructed. The coarse spaces are constructed over the different
polynomial orders and refinement levels of the immersed discretization.
Elementwise and patchwise additive Schwarz smoothing techniques are used to
mitigate the influence of the cut cells leading to convergence rates that are
independent of the cut configuration, mesh size and in certain scenarios even
the polynomial order. The multigrid approach is applied to second-order
problems arising from the Poisson equation and linear elasticity. A series of
numerical examples demonstrate the applicability of the scheme for solving
large immersed systems with multiple millions and even billions of unknowns on
massively parallel machines.
