Schur complement Domain Decomposition Methods for the solution of multiple scattering problems

@article{Pedneault2016SchurCD,
  title={Schur complement Domain Decomposition Methods for the solution of multiple scattering problems},
  author={Michel Pedneault and Catalin Turc and Yassine Boubendir},
  journal={arXiv: Numerical Analysis},
  year={2016}
}
We present a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) we connect the solutions of the subproblems via Robin boundary conditions… Expand

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References

SHOWING 1-10 OF 46 REFERENCES
A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media
TLDR
Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 min on a desktop workstation. Expand
A Multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- toExpand
Well conditioned boundary integral equations for two-dimensional sound-hard scattering problems in domains with corners
We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions inExpand
Optimized Schwarz Methods without Overlap for the Helmholtz Equation
TLDR
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. Expand
The method of polarized traces for the 2D Helmholtz equation
We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O ( N L ) , where N is the number of volumeExpand
Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem
TLDR
A preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem is presented, which has the advantage that it can be implemented as a matrix-free routine, with no additional preprocessing. Expand
A high-order integral solver for scalar problems of diffraction by screens and apertures in three-dimensional space
We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three-dimensional space. Our approach relies on new integral formulations as well asExpand
Well-posed boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains
We present a comparison between the performance of solvers based on Nystrom discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems inExpand
Wave-number estimates for regularized combined field boundary integral operators in acoustic scattering problems with Neumann boundary conditions
We study the coercivity properties and the norm dependence on the wavenumber k of certain regularized combined field boundary integral operators that we recently introduced for the solution of twoExpand
Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems
This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smoothExpand
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4
5
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