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 takes advantage of the hierarchical 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 transitioning between multigrid levels only involves the inclusion or exclusion of specific basis functions. All restriction and prolongation operators, therefore, reduce to binary matrices that do not need to be explicitly assembled or applied, saving computational time and memory. Elementwise and patchwise additive Schwarz smoothing techniques are used to mitigate the influence of the cut cells on the conditioning of the linear systems, while maintaining the parallelizability of the solver. The effectiveness of the scheme is numerically verified in various examples and convergence rates that are independent of the cut configuration, mesh size, refinement level and, in certain scenarios, even the polynomial order are shown. 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.

The finite cell method is a flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a wide range of geometrical models can be performed. The application of the finite cell method, and other immersed methods, to large real-life and industrial problems is often limited due to the conditioning problems associated with these methods. These conditioning problems have caused researchers to resort to direct solution methods. This significantly limits the maximum size of solvable systems. Iterative solvers are better suited for large-scale computations than their direct counterparts due to their lower memory requirements and suitability for parallel computing. These benefits can, however, only be exploited when systems are properly conditioned. In this contribution we present an Additive-Schwarz type preconditioner that enables efficient and parallel scalable iterative solutions of large-scale multi-level hp-refined finite cell systems.