
Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems
The Restricted Additive Schwarz method with impedance transmission condi...
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A Fixed Point Theorem for Iterative Random Contraction Operators over Banach Spaces
Consider a contraction operator T over a Banach space X with a fixed po...
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Robust treatment of cross points in Optimized Schwarz Methods
In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM...
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Analysis of the SORAS domain decomposition preconditioner for nonselfadjoint or indefinite problems
We analyze the convergence of the onelevel overlapping domain decomposi...
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On the Iteration Complexity of Hypergradient Computation
We study a general class of bilevel problems, consisting in the minimiza...
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Analysis of the Schwarz domain decomposition method for the conductorlike screening continuum model
We study the Schwarz overlapping domain decomposition method applied to ...
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On the Scalability of the Parallel Schwarz Method in OneDimension
In contrast with classical Schwarz theory, recent results in computation...
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Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation
We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the subdomain problems satisfy firstorder absorbing (impedance) transmission conditions, and exchange of information between subdomains is achieved using a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1,2,3) that it is welldefined in a tensor product of appropriate local function spaces, each with L^2 impedance boundary data. Given this, we then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedancetoimpedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2d, for rectangular domains and stripwise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedancetoimpedance maps which ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. These results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedancetoimpedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graphpartitioning software.
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