Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

中文翻译：

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用于浸入式有限元方法和浸入式等几何分析的多重网格求解器

系统矩阵的病态是浸入式有限元方法和修整等几何分析中众所周知的并发症。与物理域相交较小的元素会在系统矩阵中产生有问题的特征值，这通常会降低迭代求解器的效率和稳健性。在这篇文章中，我们研究了由 Schwarz 型方法处理的浸入式有限元系统的光谱特性，以确定这些系统在多重网格方法中作为平滑器的适用性。基于这项研究，我们开发了一种用于浸入式有限元方法的几何多重网格预处理器，它提供了与网格无关和与切割元素无关的收敛速度。这种预处理技术适用于高阶离散化，并且能够以与自由度数成线性比例的计算成本来求解大型浸入式系统。预处理器的性能在传统的拉格朗日基函数和具有均匀 B 样条和基于截断分层 B 样条的局部改进近似的等几何离散化中得到了证明。