Overview
This study examines the convergence properties of overlapping Schwarz domain decomposition methods when applied to the Helmholtz equation. The focus is on problems characterized by a large, real wavenumber and a smooth, variable wave speed in any dimension. A Cartesian perfectly-matched layer (PML) is utilized to approximate the radiation condition.
The domain decomposition strategy involves overlapping hyperrectangular subdomains, each incorporating Cartesian PMLs at its boundaries. A key aspect of the method's design is that the overlap widths between subdomains and the widths of the PMLs are set to be independent of the wavenumber.
Research Context
The Helmholtz equation presents significant computational challenges, particularly in scenarios with large wavenumbers, which often arise in wave propagation simulations. Domain decomposition methods, such as Schwarz methods, are commonly employed to break down large problems into smaller, more manageable subproblems that can be solved in parallel.
Prior work on the convergence of these methods for the Helmholtz equation, especially with non-trivial scatterers like variable wave speeds, has lacked wavenumber-explicit results. This study aims to fill that gap by providing quantitative statements about convergence that directly account for the wavenumber's influence.
Approach
The researchers investigated both parallel (additive) and sequential (multiplicative) variants of the overlapping Schwarz methods. The methods incorporate Cartesian perfectly-matched layers (PMLs) within the subdomains and at their boundaries to handle the radiation condition.
The theoretical analysis focused on determining the number of iterations required for convergence and the nature of the resulting error. This analysis considered the behavior of geometric-optic rays within the problem domain.
Numerical experiments were conducted to illustrate the derived theoretical results, providing empirical support for the convergence properties observed.
Findings
- For both parallel (additive) and sequential (multiplicative) overlapping Schwarz methods, it was demonstrated that after a specified number of iterations, the resulting error is smooth.
- The error achieved is smaller than any negative power of the wavenumber.
- The specified number of iterations depends on the behavior of geometric-optic rays within the problem domain.
- Specifically for the parallel method, the required number of iterations is less than the maximum number of subdomains that a geometric-optic ray can intersect, considering their multiplicity.
- These findings represent the first wavenumber-explicit results regarding the convergence of overlapping Schwarz methods for the Helmholtz equation.
- The study also provides the first wavenumber-explicit results on the convergence of any domain-decomposition method for the Helmholtz equation that incorporates a non-trivial scatterer, identified as a variable wave speed in this context.