The multilevel Schwarz preconditioner is one of the most popular parallel preconditioners for enhancing convergence and improving parallel efficiency. However, its parallel implementation on arbitrary unstructured triangular/tetrahedral meshes remains challenging. The challenges mainly arise from the inability to ensure that mesh hierarchies are nested, which complicates parallelization efforts. This paper systematically investigates the non-nested unstructured case of parallel multilevel algorithms and develops a highly parallel non-nested multilevel smoothed Schwarz preconditioner. The proposed multilevel preconditioner incorporates two key techniques. The first is a new parallel coarsening algorithm that preserves the geometric features of the computational domain. The second is a corresponding parallel non-nested interpolation method designed for non-nested mesh hierarchies. This new preconditioner is applied to a broad range of linear parametric problems, benefiting from the reusability of the same coarse mesh hierarchy for problems with different parameters. Several numerical experiments validate the outstanding convergence and parallel efficiency of the proposed preconditioner, demonstrating effective scalability up to 1,000 processors.

翻译：多层Schwarz预条件子是最流行的并行预条件子之一，用于加速收敛并提升并行效率。然而，其在任意非结构化三角形/四面体网格上的并行实现仍然具有挑战性。这些挑战主要源于无法确保网格层次结构具有嵌套性，从而使得并行化工作变得复杂。本文系统研究了并行多层算法在非嵌套非结构化情形下的应用，并开发了一种高度并行的非嵌套多层光滑Schwarz预条件子。所提出的多层预条件子包含两项关键技术：其一是能够保持计算区域几何特征的新型并行粗化算法；其二是针对非嵌套网格层次结构设计的相应并行非嵌套插值方法。得益于同一粗网格层次结构在不同参数问题中的可重用性，该新型预条件子被应用于广泛的线性参数化问题。若干数值实验验证了所提出预条件子优异的收敛性与并行效率，证明了其在多达1,000个处理器上的有效可扩展性。