Recursive Sweeping Preconditioner for the 3D Helmholtz Equation

This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the non-recursive 3D sweeping preconditioner, the setup cost of this new approach drops from $O(N^{4/3})$ to $O(N)$, the application cost per iteration drops from $O(N\log N)$ to $O(N)$, and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the non-recursive sweeping preconditioner to demonstrate the efficiency of the new approach.