Efficient Parallel and Sequential Domain Decomposition Methods for High-Frequency Helmholtz Problems with Variable Wave Speed

Core Concepts

The authors develop and analyze parallel and sequential overlapping Schwarz methods for solving high-frequency Helmholtz problems with variable wave speed, using Cartesian perfectly-matched layers (PML) at the subdomain boundaries. They prove that the error decays smoothly and exponentially quickly in the number of iterations, with the number of iterations depending on the behavior of the geometric-optic rays.

Abstract

The content presents a rigorous analysis of parallel and sequential overlapping Schwarz methods for solving high-frequency Helmholtz problems with variable wave speed, using Cartesian PML at the subdomain boundaries. Key highlights: The authors consider Helmholtz problems in any dimension with large, real wavenumber and smooth variable wave speed, with the radiation condition approximated by a Cartesian PML. For both parallel (additive) and sequential (multiplicative) overlapping Schwarz methods, they show that after a specified number of iterations, the error is smooth and smaller than any negative power of the wavenumber. The number of iterations depends on the behavior of the geometric-optic rays, with the parallel method requiring less iterations than the maximum number of subdomains that a ray can intersect. These are the first wavenumber-explicit results about convergence of overlapping Schwarz methods for the Helmholtz equation, and the first such results for any domain-decomposition method with a non-trivial scatterer (variable wave speed). The authors make clear the key properties of PML that are required to obtain these results, and discuss how the results extend to other complex absorption operators. The results are valid on fixed domains for sufficiently large wavenumber, with the PML widths and domain decomposition overlaps independent of the wavenumber.

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Key Insights Distilled From

Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems

by Jeffrey Galk... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.02156.pdf

Deeper Inquiries

How can the results be extended to obtain convergence rates that are also explicit in the geometric parameters of the domain decomposition, such as the PML widths and subdomain overlaps

To extend the results to obtain convergence rates explicit in the geometric parameters of the domain decomposition, such as the PML widths and subdomain overlaps, additional technical arguments would be required. One approach could involve analyzing the impact of varying PML widths and subdomain overlaps on the error propagation matrix T. By studying how these geometric parameters affect the error propagation through the iterations, it may be possible to derive convergence rates that explicitly account for these parameters. This would involve a more detailed analysis of how the error travels between subdomains and how the PML approximations influence the convergence behavior.

What are the implications of these results for the practical performance and implementation of overlapping Schwarz methods for high-frequency Helmholtz problems with variable wave speed

The results obtained in this work have significant implications for the practical performance and implementation of overlapping Schwarz methods for high-frequency Helmholtz problems with variable wave speed. By showing convergence rates that are smooth and smaller than any negative power of the wavenumber, the study provides valuable insights into the behavior of these methods in the context of nontrapping Helmholtz problems. This information can guide the optimization of domain decomposition strategies for efficiently solving such problems, especially when dealing with large, real wavenumbers and smooth variable wave speeds. Implementing these methods with a better understanding of their convergence properties can lead to more effective and accurate numerical simulations of wave propagation phenomena.

Can the techniques developed in this work be applied to analyze other types of domain decomposition methods, such as non-overlapping methods, for high-frequency Helmholtz problems with variable coefficients

While the techniques developed in this work are specifically tailored for overlapping Schwarz methods applied to high-frequency Helmholtz problems with variable coefficients, they can potentially be adapted to analyze other types of domain decomposition methods as well. For instance, the approach of studying error propagation matrices and trajectories of the flow associated with the differential operator could be extended to non-overlapping methods. By investigating how the error propagates through non-overlapping subdomains and how the boundary conditions influence convergence, similar convergence results could potentially be derived for non-overlapping domain decomposition methods in the context of high-frequency Helmholtz problems with variable coefficients. This would require adapting the analysis to the specific characteristics and constraints of non-overlapping methods.