The Hierarchical Poincare-Steklov (HPS) solver for elliptic PDEs: A tutorial

2015/06/03 by Per‐Gunnar Martinsson, Martinsson, P. G. · 5 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1506.01308

openalex publication_date 2015/06/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct solver with O(N^1.5) complexity for the pre-computation and O(N log N) complexity for the solve. The fact that the solver is direct is a principal feature of the scheme, and makes it particularly well suited to solving problems for which iterative solvers struggle; in particular for problems with highly oscillatory solutions. This note is intended as a tutorial description of the scheme, and draws heavily on previously published material.

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