Shifted HSS solvers for the indefinite Helmholtz equation

TL;DR

This paper introduces a scalable iterative method for solving the indefinite Helmholtz equation using shifted HSS iterations, suitable for high-performance computing, with proven robustness and verified numerical results.

Contribution

The paper presents a novel shifted HSS iterative solver for the indefinite Helmholtz equation that is robust with respect to mesh size and wave number, and optimized for parallel high-performance computing.

Findings

Iteration is k- and mesh-robust with O(k) iterations.

Solver is suitable for parallel implementation with O(N) processors.

Numerical results confirm theoretical convergence and scalability.

Abstract

We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results in both 2D and 3D verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.