An iterative solver for the 3D Helmholtz equation

We develop a frequency–domain iterative solver for numerical simulation of acoustic waves in 3D heterogeneous media. It is based on the application of a unique preconditioner to the Helmholtz equation that ensures convergence for Krylov subspace iteration methods. Effective inversion of the preconditioner involves the Fast Fourier Transform (FFT) and numerical solution of a series of boundary value problems for ordinary differential equations. Matrix-by-vector multiplication for iterative inversion of the preconditioned matrix involves inversion of the preconditioner and pointwise multiplication of grid functions. Our solver has been verified by benchmarking against exact solutions and a time-domain solver. •A frequency–domain iterative solver for the 3D Helmholtz equation is proposed.•A preconditioner is the shifted Laplacian for a specially constructed layered model.•Using the FFT is a core item to invert the preconditioner.•Second-level preconditioning helps to improve convergence of the method.