An iterative solver for the HPS discretization applied to three dimensional Helmholtz problems

TL;DR

This paper introduces the first efficient iterative solver for the HPS discretization of 3D Helmholtz problems, significantly improving computational efficiency for large-scale, high-frequency simulations.

Contribution

It develops an iterative GMRES-based solver with a block-Jacobi preconditioner for HPS discretizations, enabling matrix-free solutions for large 3D Helmholtz problems.

Findings

Solver handles problems with over a billion unknowns

Achieves 4-digit accuracy in under 20 minutes for large-scale problems

Demonstrates superior performance over direct solvers in large 3D cases

Abstract

This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve.