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… 

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References

SHOWING 1-10 OF 28 REFERENCES

Optimized Schwarz Methods without Overlap for the Helmholtz Equation

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.

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

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

  • M. Gander
  • Computer Science
    SIAM 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

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.

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

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

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.