Shifted HSS solvers for the indefinite Helmholtz equation
This provides an efficient solution for large-scale indefinite Helmholtz problems in computational physics, enabling use of high-performance computing systems, though it is incremental as it builds on existing HSS and multigrid methods.
The paper tackles solving the indefinite Helmholtz equation by developing an iterative solver based on a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to a shifted operator, proving it is k- and mesh-robust with O(k) iterations and converges in O(k) wallclock time using multigrid approximations on parallel processors.
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.