Towards modular Hierarchical Poincaré-Steklov solvers

2025/10/30 by Outrata, Michal, Lorca, José Pablo Lucero · 2 citations
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2510.26945

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.

Citations

• Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers
• A Two-Level Direct Solver for the Hierarchical Poincaré-Steklov Method
• A Fast Direct Solver for Elliptic PDEs on a Hierarchy of Adaptively Refined Quadtrees
• A high-order fast direct solver for surface PDEs
• An iterative solver for the HPS discretization applied to three dimensional Helmholtz problems
• Robust treatment of cross points in Optimized Schwarz Methods
• A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems
• An adaptive high order direct solution technique for elliptic boundary value problems
• An accelerated Poisson solver based on multidomain spectral\n discretization
• The Hierarchical Poincare-Steklov (HPS) solver for elliptic PDEs: A tutorial
• Cross-Points in Domain Decomposition Methods with a Finite Element Discretization
• A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media
• A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method

Cited by

• On the Spectral Clustering in Algebraic Multigrid Methods
• Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers

Related

• Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers
• A Two-Level Direct Solver for the Hierarchical Poincaré-Steklov Method
• The Hierarchical Poincare-Steklov (HPS) solver for elliptic PDEs: A tutorial
• Exponential convergence of the hp Virtual Element Method with corner singularities
• A Fast Direct Solver for Elliptic PDEs on a Hierarchy of Adaptively Refined Quadtrees