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

翻译：浸入式有限元方法被开发为一种规避传统有限元分析中高代价网格生成的手段。然而，此类方法中所得线性方程组的数值病态性对迭代求解器构成了挑战。本研究聚焦于采用自适应求积、自适应网格细化（AMR）及弱施加边界条件的Nitsche方法的有限胞元法（FCM）。针对离散化问题，我们采用自适应几何多重网格求解器。我们研究了Nitsche方法中网格依赖的稳定化参数对几何多重网格求解器性能的影响，及其对多层级配置的普遍意义。基于广义特征值问题的全局与局部估计被用于选取稳定化参数。研究发现，求解器的收敛速率显著受稳定化参数、估计方式的选择以及多层级配置中稳定化参数的处理方式影响。在各网格上计算的局部估计被证明是一种稳健方法，能够促使几何多重网格求解器快速收敛。