# Optimized Schwarz Methods with Overlap for the Helmholtz Equation

@article{Gander2016OptimizedSM, title={Optimized Schwarz Methods with Overlap for the Helmholtz Equation}, author={Martin J. Gander and Hui Zhang}, journal={SIAM J. Sci. Comput.}, year={2016}, volume={38} }

Optimized Schwarz methods are based on optimized transmission conditions between subdomains and can have substantially improved convergence behavior compared to classical Schwarz methods. This is especially true when the method is applied to the Helmholtz equation, and better transmission conditions in form of perfectly matched layers have for example led to the new class of sweeping preconditioners. We present here for the first time a complete analysis of optimized Schwarz methods with…

## 28 Citations

### Natural Domain Decomposition Algorithms for the Solution of Time-Harmonic Elastic Waves

- Computer ScienceSIAM J. Sci. Comput.
- 2020

This work proves that the classical Schwarz method is not convergent when applied to the Navier equations, and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method, and introduces a new Schwarz method with adapted transmission conditions.

### The influence of domain truncation on the performance of optimized Schwarz methods

- Computer Science
- 2018

It is proved here rigorously for a two-subdomain decomposition that the asymptotic performance of optimized Schwarz methods derived from an unbounded domain analysis still holds in the case of a bounded domain, but the constants in the best choice of parameters and convergence rate estimates are influenced by the domain truncation.

### Closed form optimized transmission conditions for complex diffusion with many subdomains

- Computer ScienceArXiv
- 2022

This work investigates the optimization of transmission conditions directly for the many subdomain case, and studies the optimization problem in the limit when the number of subdomains goes to inﬁnity, using the tool of limiting spectra.

### The method of polarized traces for the 2D Helmholtz equation

- Computer ScienceJ. Comput. Phys.
- 2016

We have developed a fast solver for the 3D Helmholtz equation, in heterogeneous, constant density, acoustic media, in the high-frequency regime. The solver is based on the method of polarized traces,…

### A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

- Computer ScienceSIAM Rev.
- 2019

The goal of the present manuscript is to show that this class of preconditioners developed by researchers with various backgrounds are based on a common mathematical principle, and they can all be formulated in the context of domain decomposition methods called optimized Schwarz methods.

### AN EFFICIENT TRANSMISSION OPERATOR FOR COMPUTING WAVE PROPAGATION BY DOMAIN DECOMPOSITION

- Computer Science, MathematicsProceedings of the 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015)
- 2019

Transmission operators approximating the Dirichlet to Neumann (DtN) operator which is known to be near optimal are proposed which can be used to solve wave propagation problems such as the Helmholtz equation using only the solution of problems involving sparse matrices.

### Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2021

### Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

- Computer ScienceMath. Comput.
- 2019

The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption.

### Domain decomposition preconditioning for the Helmholtz equation

- Mathematics, Computer Science
- 2015

A two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation and a coarse space that is based on local eigenproblems involving the Dirichlet-toNeumann operator is developed.

### Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

- Computer Science, MathematicsETNA - Electronic Transactions on Numerical Analysis
- 2021

It is proved that the Faber preconditioner is nonsingular for degrees one and two of the truncated series, which reduces the number of GMRES iterations.

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