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

@article{Gander2007AnOS, title={An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation}, author={Martin J. Gander and Laurence Halpern and Fr{\'e}d{\'e}ric Magoul{\`e}s}, journal={International Journal for Numerical Methods in Fluids}, year={2007}, volume={55} }

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring…

## 107 Citations

### Optimized Schwarz Methods with Overlap for the Helmholtz Equation

- Computer ScienceSIAM J. Sci. Comput.
- 2016

This work presents here for the first time a complete analysis of optimized Schwarz methods with overlap for the Helmholtz equation, and obtains closed form asymptotically optimized transmission conditions for the case of two subdomains.

### 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.

### Optimized double sweep Schwarz method by complete radiation boundary conditions

- Computer ScienceComput. Math. Appl.
- 2016

### Optimized Schwarz Methods for Maxwell's Equations

- Computer ScienceSIAM J. Sci. Comput.
- 2009

It is shown here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation.

### Heterogeneous Optimized Schwarz Methods for Coupling Helmholtz and Laplace Equations

- Computer Science
- 2017

Zeroth order optimized transmission conditions for the coupling between the Helmholtz equation and the Laplace equation are proposed, giving asymptotically optimized choices for the parameters, and illustrating the analytical results with numerical experiments.

### Restrictions on the Use of Sweeping Type Preconditioners for Helmholtz Problems

- Computer Science
- 2017

This short manuscript gives a simple, compact representation of sweeping type preconditioners, and finds that iteration numbers of all these methods remain robust for very small contrast variations, but then deteriorate, with linear growth both in the wave number as well as in the number of subdomains, as soon as the contrast variations reach order one.

### Optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients

- Mathematics, Computer Science
- 2008

A global-in-time non-overlapping Schwarz method applied to the one dimensional unsteady diffusion equation and derives a new approach to determine the convergence factor, which enables to optimize the convergence speed.

### A unified theory of non-overlapping Robin-Schwarz methods - continuous and discrete, including cross points

- MathematicsArXiv
- 2022

. Non-overlapping Schwarz methods with generalized Robin transmission conditions were originally introduced by B. Despr´es for time-harmonic wave propagation problems and have largely developed over…

### Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients

- Computer Science
- 2014

It is shown that the classical Schwarz algorithm is always divergent when coefficient jumps are present along the interface, and optimized Schwarz methods are developed which can take coefficient jumps into account in their transmission conditions.

### Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

- Computer ScienceSIAM J. Numer. Anal.
- 2015

A nonoverlapping optimized Schwarz algorithm with a high-order transmission condition for the Helmholtz equation posed in waveguides is presented and damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the Schwarz algorithm.

## References

SHOWING 1-10 OF 28 REFERENCES

### 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.

### Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

- Computer Science
- 2004

### 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.

### Optimized Schwarz Methods

- Computer ScienceSIAM J. Numer. Anal.
- 2006

This paper analyzes these new methods for symmetric positive definite problems and shows their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.

### A non Overlapping Domain Decomposition Method for the Exterior Helmholtz Problem

- Computer Science
- 1997

An alternative domain decomposition algorithm that is better suited for the exterior Helmholtz problem is presented, in a formalism that can use either one or two Lagrange multiplier fields for solving the corresponding interface problem by a Krylov method.

### Non-overlapping Schwarz methods with optimized transmission conditions for the H

- Computer Science
- 2004

### Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems

- Computer Science
- 2000

### Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers?

- MathematicsSIAM Rev.
- 2000

It is proved that, in two and more space dimensions, it is impossible to eliminate the so-called pollution effect of the Galerkin FEM.

### Schwarz Alternating Method

- Computer Science
- 1998

A discrete technique of the Schwarz alternating method is presented, to combine the Ritz-Galerkin and finite element methods, well suited for solving singularity problems in parallel.