Abstract

The indefinite Helmholtz equation, obtained by Fourier transformation of the wave equation in time, arises in many applications including acoustics, elasticity, electromagnetism, geophysics, and quantum mechanics. Scalable iterative solvers for discretisations of the indefinite Helmholtz equation remain a challenging problem in scientific computing and numerical analysis. I will present such an iterative solver that combines shift preconditioning, Hermitian/skew-Hermitian splitting and multigrid methods. Standard multigrid methods can be used with local smoothers (such as Jacobi smoothers) that can be parallelised by domain decomposition with minimal overlaps, unlike some other solver approaches for the indefinite Helmholtz equation. I will present a proof that the solver converges at a rate that is independent of the frequency k, the mesh size h, and all other parameters of the problem, provided that O(k) inner iterations are performed. I will also present numerical experiments that confirm this result.