We revisit the Hierarchical Poincar\'e--Steklov (HPS) method within a preconditioned iterative framework. Originally introduced as a direct solver for elliptic boundary-value problems, the HPS method combines nested dissection with tensor-product spectral element discretizations, even though it has been shown in other contexts[8]. Building on the iterative variant proposed in[1], we reinterpret the hierarchical merge structure of HPS as a natural multigrid preconditioner. This perspective unifies direct and iterative formulations of HPS connecting it to multigrid domain decomposition. The resulting formulation preserves the high accuracy of spectral discretizations while enabling flexible iterative solution strategies. Numerical experiments in two dimensions demonstrate the performance and convergence behavior of the proposed approach.

翻译：我们在预条件迭代框架下重新审视层次化Poincaré–Steklov（HPS）方法。该方法最初作为椭圆型边值问题的直接求解器提出，结合了嵌套剖分与张量积谱元离散化，尽管在其他文献[8]中已展示其应用。基于文献[1]提出的迭代变体，我们将HPS的层次化合并结构重新解释为一种自然的多重网格预条件子。这一视角统一了HPS的直接与迭代形式，并将其与多重网格区域分解方法相联系。所得公式在保持谱离散化高精度的同时，支持灵活的迭代求解策略。二维数值实验验证了所提方法的性能与收敛特性。