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Immersed finite element methods have been developed as a means to circumventthe costly mesh generation required in conventional finite element analysis.However, the numerical ill-conditioning of the resultant linear system ofequations in such methods poses a challenge for iterative solvers. In thiswork, we focus on the finite cell method (FCM) with adaptive quadrature,adaptive mesh refinement (AMR) and Nitsche's method for the weak imposition ofboundary conditions. An adaptive geometric multigrid solver is employed for thediscretized problem. We study the influence of the mesh-dependent stabilizationparameter in Nitsche's method on the performance of the geometric multigridsolver and its implications for the multilevel setup in general. A global and alocal estimate based on generalized eigenvalue problems are used to choose thestabilization parameter. We find that the convergence rate of the solver issignificantly affected by the stabilization parameter, the choice of theestimate and how the stabilization parameter is handled in multilevelconfigurations. The local estimate, computed on each grid, is found to be arobust method and leads to rapid convergence of the geometric multigrid solver.

The Influence of Nitsche Stabilization on Geometric Multigrid for the Finite Cell Method