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Does the Partition of Unity Influence the Convergence of Schwarz Methods?

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Domain Decomposition Methods in Science and Engineering XXV (DD 2018)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 138))

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Abstract

The classical alternating Schwarz method does not need a partition of unity in its definition [3]: one solves one subdomain after the other, stores subdomain solutions, and always uses the newest data available from the neighboring subdomains. In the parallel Schwarz method introduced by Lions in [10], where all subdomains are solved simultaneously, one also stores subdomain solutions, but one has to distinguish two cases: if in the decomposition there are never more than two subdomains that intersect, which we call the no crosspoint assumption, then one also does not need a partition of unity to define the method, one simply takes data from the neighboring subdomains with which the subdomain intersects, and in that case the parallel Schwarz method has a variational interpretation [10].

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Correspondence to Martin J. Gander .

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Gander, M.J. (2020). Does the Partition of Unity Influence the Convergence of Schwarz Methods?. In: Haynes, R., et al. Domain Decomposition Methods in Science and Engineering XXV. DD 2018. Lecture Notes in Computational Science and Engineering, vol 138. Springer, Cham. https://doi.org/10.1007/978-3-030-56750-7_1

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