Additive Sweeping Preconditioner for the Helmholtz Equation

TL;DR

This paper presents a novel additive sweeping preconditioner for the Helmholtz equation that significantly reduces frequency dependence in iterative solutions by dividing the domain into layers and optimizing boundary transmission conditions.

Contribution

The paper introduces a new additive sweeping preconditioner based on PML and boundary value focus, improving the efficiency of Helmholtz equation solvers especially at high frequencies.

Findings

Iteration count is nearly independent of frequency.

Numerical examples demonstrate high efficiency.

Effective approximation of solution operator decomposition.

Abstract

We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the intermediate waves. This approach can be viewed as an effective approximation of an additive decomposition of the solution operator. When combined with the standard GMRES solver, the iteration number is essentially independent of the frequency. Several numerical examples are tested to show the efficiency of this new approach.