We revisit the Hierarchical Poincar\'{e}-Steklov (HPS) method for the Poisson equation using standard Q1 finite elements, building on the original in work on HPS of Martinsson from 2013. While corner degrees of freedom were implicitly handled in that work, subsequent spectral-element implementations have typically avoided them. In Q1-FEM, however, corner coupling cannot be factored out, and we show how the HPS merge procedure naturally accommodates it when corners are enclosed by elements. This clarification bridges a conceptual gap between algebraic Schur-complement methods and operator-based formulations, providing a consistent path for the FEM community to adopt HPS to retain the Poincar\'{e}-Steklov interpretation at both continuous and discrete levels.

翻译：我们基于Martinsson于2013年关于分层泊松-斯捷克洛夫方法的原始工作，重新审视了使用标准Q1有限元求解泊松方程的分层泊松-斯捷克洛夫方法。虽然该原始工作中隐含处理了角点自由度，但后续的谱元实现通常回避了它们。然而在Q1有限元中，角点耦合无法被分解，我们展示了当角点被单元包围时，HPS合并过程如何自然地容纳这一耦合。这一阐释弥合了代数舒尔补方法与基于算子的表述之间的概念鸿沟，为有限元社区采用HPS提供了一条连贯的路径，以在连续和离散层面均保持泊松-斯捷克洛夫解释。