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This manuscript presents an efficient solver for the linear system thatarises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of threedimensional variable coefficient Helmholtz problems. Previous work on the HPSmethod has tied it with a direct solver. This work is the first efficientiterative solver for the linear system that results from the HPSdiscretization. The solution technique utilizes GMRES coupled with a locallyhomogenized block-Jacobi preconditioner. The local nature of the discretizationand preconditioner naturally yield the matrix-free application of the linearsystem. Numerical results illustrate the performance of the solution technique.This includes an experiment where a problem approximately 100 wavelengths ineach direction that requires more than a billion unknowns to achieveapproximately 4 digits of accuracy takes less than 20 minutes to solve.

An iterative solver for the HPS discretization applied to three dimensional Helmholtz problems