Optimized Schwarz Methods with Overlap for the Helmholtz Equation

By Martin J. Gander, Hui Zhang. SIAM Journal on Scientific Computing, 2016

403 citations recorded by the source corpus.

Optimized Schwarz methods are based on optimized transmission conditions between subdomains and can have substantially improved convergence behavior compared to classical Schwarz methods. This is especially true when the method is applied to the Helmholtz equation, and better transmission conditions in form of perfectly matched layers have for example led to the new class of sweeping preconditioners. We present here for the first time a complete analysis of optimized Schwarz methods with overlap for the Helmholtz equation. We obtain closed form asymptotically optimized transmission conditions for the case of two subdomains, and study numerically the influence of the number of subdomains on this optimized choice.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/ba9af3ea1157e72dd94dcb3409c79310d63346d6

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In Atomic and Molecular Physics, and Optics

In Computational Mechanics

In Electrical and Electronic Engineering

Citations in the corpus

Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method (2005, methodology): “are still less efficient than those obtained with a continuous approach, see Gander et al. [2002], but the implementation of the previous method doesn’t depends on a priori knowledge of the problem to be solved (coefficients of the”
Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods (2012, methodology): “In this case, we can use Fourier series in the direction of the interface to explicitly compute how each Fourier mode converges, see for example [34].”
Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems (2003, methodology): “We base our approach on an analysis of the algorithm at the continuous level, in the spirit of [11, 12] and [10]; see also the basic reference [6].”
Lagrangian formulation of domain decomposition methods: A unified theory (2006, neutral): “And for the non-overlapping Schwarz method [20,21,33–35].”
Optimized Schwarz Methods (2006, methodology): “The optimization problem for the Robin transmission conditions was then completely solved for this case in [22] and a simple strategy to optimize the second order transmission conditions was also presented.”
A review of finite-element methods for time-harmonic acoustics (2006, methodology): “Optimized coeﬃcients have been chosen to minimize the convergence rate of the Jacobi algorithm in the closely related additive Schwarz method with no overlap [72].”
A FEM domain decomposition method for photonic and electromagnetic band gap structures (2006, methodology): “This would lead to the Dirichlet-to-Dirichlet (DtD) or Dirichlet-to-Neumann (DtN) DDM algorithms that converge only for elliptic-type problems [13].”