Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincar\'e-Steklov Solvers

TL;DR

This paper reinterprets the Hierarchical Poincaré-Steklov method as a multigrid preconditioner for Helmholtz problems, aiming to reduce memory use by replacing the coarse solve with iterative methods.

Contribution

It introduces a flexible multigrid preconditioning framework for HPS, enabling tunable trade-offs between memory and computational time.

Findings

Numerical experiments demonstrate effective trade-offs between memory and solution time.

The preconditioned approach improves efficiency over unpreconditioned GMRES.

Replacing the coarse solve with iterative methods maintains accuracy while reducing memory usage.

Abstract

We revisit the Hierarchical Poincar\'e-Steklov (HPS) method in a preconditioned iterative setting for variable-coefficient Helmholtz problems with impedance boundary conditions. HPS is commonly presented as a direct solver based on nested dissection and high-order tensor-product discretizations; here we recast its hierarchical merge tree as a multilevel preconditioner for the assembled skeleton (trace) system. The main goal is to flexibilize the final, memory-intensive coarse stage of direct HPS by replacing the exact coarse solve with a small, fixed amount of iterative work, thereby exposing tunable trade-offs between memory footprint and time to solution. Numerical experiments on a two-dimensional scattering benchmark illustrate these trade-offs and compare against both unpreconditioned GMRES and the classic direct HPS pipeline with an exact coarse space.