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

This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with an exact block-Jacobi preconditioner. The construction of the block-Jacobi preconditioner involves two nested local solves that are accelerated by local homogenization. The local nature of the discretization and preconditioner naturally yield matrix-free application of the linear system. A distributed memory implementation allows the solution technique to tackle problems approximately $50$ wavelengths in each direction requiring more than a billion unknowns to get approximately 7 digits of accuracy in less than an hour. Additional numerical results illustrate the performance of the solution technique.