Shifted HSS solvers for the indefinite Helmholtz equation

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.

Figures (3)

• Figure 1: Solution of the uniform source case when $k=160$.
• Figure 2: Solution of the box source case when $k=160$.
• Figure 3: Weak scaling for the primal formulation in 2D (\ref{['fig:timing-2D']}) and 3D (\ref{['fig:timing-3D']}) 3D.

Theorems & Definitions (26)

• Definition 3.1: $\gamma=k$ HSS iteration in primal form
• Proposition 3.2: HSS convergence for the shifted primal system
• Proof 1
• Corollary 3.3
• Proof 2
• Definition 3.4: $\gamma=k$ HSS iteration in mixed form
• Proposition 3.5: HSS convergence for the shifted mixed system
• Proof 3
• Corollary 3.6
• Proof 4