Towards modular Hierarchical Poincar\'{e}-Steklov solvers

TL;DR

This paper revisits the Hierarchical Poincaré-Steklov method for solving the Poisson equation with Q1 finite elements, clarifying corner coupling handling and bridging algebraic and operator-based approaches.

Contribution

It clarifies how to incorporate corner degrees of freedom in the HPS method for FEM, enabling broader adoption and consistency with Poincaré-Steklov theory.

Findings

HPS merge procedure naturally handles corner coupling.

Bridges gap between algebraic and operator-based methods.

Facilitates FEM community adoption of HPS

Abstract

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.