Optimized Schwarz Methods without Overlap for the Helmholtz Equation

@article{Gander2002OptimizedSM,
  title={Optimized Schwarz Methods without Overlap for the Helmholtz Equation},
  author={Martin J. Gander and Fr{\'e}d{\'e}ric Magoul{\`e}s and Fr{\'e}d{\'e}ric Nataf},
  journal={SIAM J. Sci. Comput.},
  year={2002},
  volume={24},
  pages={38-60}
}
The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps… 

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References

SHOWING 1-10 OF 47 REFERENCES
Symmetrized Method with Optimized Second-Order Conditions for the Helmholtz Equation
TLDR
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.
Overlapping Schwarz Algorithms for Solving Helmholtz''s Equation
TLDR
This paper gives a progress report on the development of a new family of domain decomposition methods for the solution of Helmholtz''s equation through a sequence of discontinuous iterates based on overlapping Schwarz methods.
Schwarz Alternating Method
TLDR
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.
The optimized order 2 method : Application to convection-diffusion problems
Domain Decomposition Algorithms for Indefinite Elliptic Problems
TLDR
This paper presents an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions, and shows the rate of convergence to be independent of the number of degrees of freedom and thenumber of local problems.
Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers?
TLDR
It is proved that, in two and more space dimensions, it is impossible to eliminate the so-called pollution effect of the Galerkin FEM.
The optimized order 2 method: application to convection-diffusion problems
TLDR
This work presents an iterative, non-overlapping domain decomposition method, which permits to solve very big problems which couldn’t be solved on only one processor, and shows a very fast convergence, nearly independant both of the physical and the discretization parameters.
The Best Interface Conditions for Domain Decomposition Methods : Absorbing Boundary Conditions
We present an iterative domain decomposition method for solving the convection-diffusion equation. In order to have very fast convergence, we use differential interface conditions of order 1 in the
Domain Decomposition Methods in Science and Engineering
Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.- Hybrid Schwarz-Multigrid Methods for the Spectral
Application of a domain decomposition method with Lagrange multipliers to acoustic problems arising
TLDR
The purpose of this paper is to study the robustness and efficiency of iterative methods for the solution of the associated interface problem for three-dimensional interior problems arising from the automotive industry.
...
...