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Towards modular Hierarchical Poincaré-Steklov solvers

Michal Outrata, José Pablo Lucero Lorca

TL;DR

The paper addresses integrating the Hierarchical Poincaré–Steklov framework with standard $Q_1$ finite elements for the Poisson problem on a rectangle, clarifying how corner coupling can be incorporated while preserving the $Poincaré–Steklov$ interpretation. It proposes a modular exposition that decouples discretization from corner treatment and shows how discrete local DtN operators $S_e$ can be merged via Schur complements through a recursive skeletonization process. Key contributions include a FEM-friendly corner merge strategy within HPS, a clear separation of assembly/merge/recursion stages, and numerical demonstrations of speedups for multiple right-hand sides on modest hardware. By aligning HPS with standard FEM and operator-based viewpoints, the work aims to facilitate adoption of HPS in the FEM community and bridge algebraic and operator-based solvers, preserving the Poincaré–Steklov perspective at both continuous and discrete levels.

Abstract

We revisit the Hierarchical Poincaré-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 Poincaré-Steklov interpretation at both continuous and discrete levels.

Towards modular Hierarchical Poincaré-Steklov solvers

TL;DR

The paper addresses integrating the Hierarchical Poincaré–Steklov framework with standard finite elements for the Poisson problem on a rectangle, clarifying how corner coupling can be incorporated while preserving the interpretation. It proposes a modular exposition that decouples discretization from corner treatment and shows how discrete local DtN operators can be merged via Schur complements through a recursive skeletonization process. Key contributions include a FEM-friendly corner merge strategy within HPS, a clear separation of assembly/merge/recursion stages, and numerical demonstrations of speedups for multiple right-hand sides on modest hardware. By aligning HPS with standard FEM and operator-based viewpoints, the work aims to facilitate adoption of HPS in the FEM community and bridge algebraic and operator-based solvers, preserving the Poincaré–Steklov perspective at both continuous and discrete levels.

Abstract

We revisit the Hierarchical Poincaré-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 Poincaré-Steklov interpretation at both continuous and discrete levels.

Paper Structure

This paper contains 6 sections, 23 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The HPS method with corners
  3. The discretization & assembly stage
  4. The merge stage
  5. The recursion stage
  6. Numerical illustration

Figures (1)

  • Figure 1: Left: two neighbouring subdomains $\Omega_1,\Omega_2$ with the index sets of the grids. We have $\iota_1^I\equiv \iota_1^R$ and $\iota_2^I\equiv \iota_2^L$ and also see the corner index sets $\iota_1^C,\iota_2^C$, although not separately highlighted. Finally, we set $\iota^{\partial}_{e} := \iota^{L}_{e} \cup \iota^{R}_{e} \cup \iota^{T}_{e} \cup \iota^{B}_{e} \cup \iota^{C}_{e}$ so that $\iota_{e} = \iota^{\mathrm{int}}_{e} \cup \iota^{\partial}_{e}$ for any $e$. Right: (incomplete) illustration of the nested dissection merge hierarchy ordering, see Martinsson2013.