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

@article{Brunet2019NaturalDD, title={Natural Domain Decomposition Algorithms for the Solution of Time-Harmonic Elastic Waves}, author={Romain Brunet and Victorita Dolean and Martin J. Gander}, journal={SIAM J. Sci. Comput.}, year={2019}, volume={42}, pages={A3313-A3339} }

We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove 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. We then introduce…

## 4 Citations

### El-WaveHoltz: A Time-Domain Iterative Solver for Time-Harmonic Elastic Waves

- MathematicsComputer Methods in Applied Mechanics and Engineering
- 2022

### Can Classical Schwarz Methods for Time-harmonic Elastic Waves Converge?

- Computer ScienceLecture Notes in Computational Science and Engineering
- 2020

We show that applying a classical Schwarz method to the time harmonic Navier equations, which are an important model for linear elasticity, leads in general to a divergent method for low to…

### Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

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

It is shown here that when impedance transmission conditions are used in the case of the algorithm applied to the equations with absorption, under certain assumptions, scalability can be achieved and no coarse space is required and this result is also true for the iterative version of the method at the continuous level.

### 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.

## References

SHOWING 1-10 OF 43 REFERENCES

### Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

- Computer ScienceJ. Comput. Phys.
- 2015

### Optimized Schwarz Methods without Overlap for the Helmholtz Equation

- Computer ScienceSIAM J. Sci. Comput.
- 2002

A variant of the Schwarz method which converges without overlap for the Helmholtz equation is studied, and it is shown that the key ingredients for such an algorithm are the transmission conditions, which lead to convergence of the algorithm in a finite number of steps.

### 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.

### Can Classical Schwarz Methods for Time-harmonic Elastic Waves Converge?

- Computer ScienceLecture Notes in Computational Science and Engineering
- 2020

We show that applying a classical Schwarz method to the time harmonic Navier equations, which are an important model for linear elasticity, leads in general to a divergent method for low to…

### Symmetrized Method with Optimized Second-Order Conditions for the Helmholtz Equation

- Computer Science, Mathematics
- 1998

A schwarz type domain decomposition method for the Helmholtz equation is considered and the computational domain Ω is decomposed into N nonoverlapping subdomains, increasing the convergence speed dramatically.

### A rapidly converging domain decomposition method for the Helmholtz equation

- Computer ScienceJ. Comput. Phys.
- 2013

### 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 Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain

- Computer ScienceSIAM J. Numer. Anal.
- 2013

A domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems using the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers.

### An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

- Computer Science
- 2007

A new optimized Schwarz method without overlap in the 2d case is presented, which uses a different Robin condition for neighbouring subdomains at their common interface, and which is called two‐sided Robin condition.

### New Nonoverlapping Domain Decomposition Methods for the Harmonic Maxwell System

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2006

Fourier analysis is used to derive suitable families of transmission conditions in R3 that involve second order tangential differential operators and that guarantee convergence for both propagative and evanescent modes that depend upon parameters chosen to optimize the convergence rate of the corresponding iterative algorithm.