Recursive Sweeping Preconditioner for the Three-Dimensional Helmholtz Equation

This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in three dimensions. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers. The key idea is to apply the sweeping preconditioner recursively to the quasi-two-dimensional auxiliary problems introduced in the three-dimensional (3D) sweeping preconditioner. Compared to the nonrecursive 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 increases only mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach.