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

Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids