Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes
type: Item
instance of: scholarly article
title: Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes
zbMATH Open document ID: 0815.65127
DOI: 10.1007/BF02142697
author: Tony F. Chan
author: Jun Zou
published in: Numerical Algorithms
publication date: 1995-06-30T00:00:00Z
review text: This paper considers a number of variants of the additive Schwarz method introduced by \textit{M. Dryja} and \textit{O. B. Widlund} in 1987. The focus here is on extensions of this algorithm to unstructured meshes. The preconditioners defined by domain decomposition methods of this family use two triangulations, a coarse with characteristic mesh size \(H\), and a fine on which the finite element problem is defined. A global component of the preconditioner is constructed from a finite element problem on the coarse mesh and, in addition, potentially many local components of the preconditioner are given in terms of the fine mesh and an overlapping decomposition of the region. The theory was first developed in detail for the case when the coarse space is a subspace of the finite element space on the fine mesh, and the two meshes are quasi uniform. The generalization considered here addresses situations where the coarse mesh is selected independently of the fine mesh and where the elements are shape regular but the meshes not necessarily quasi uniform. The estimate of the condition number of a domain decomposition method of this class always involves a decomposition of the elements of the entire space into components of the subspaces and the study of the stability of this decomposition. Often, the element of the coarse mesh subspace is chosen as the \(L_ 2\) projection onto the coarse space, or in the general case, the standard finite element interpolant of that projection. This is where the requirement of quasi uniformity enters the analysis. That condition can be relaxed by using an idea, employed in approximation theory for many years, namely averaging and interpolation. This is in fact how the first good proof of the optimality of this Schwarz method was constructed; cf. \textit{M. Dryja} and \textit{O. B. Widlund}, in ``Iterative methods for large linear systems'', Academic Press (1989; Zbl 0703.68010). That proof extends immediately to any mesh with shape regular e…
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 720557
zbMATH Keywords: additive Schwarz method
zbMATH Keywords: unstructured meshes
zbMATH Keywords: preconditioners
zbMATH Keywords: domain decomposition methods
zbMATH Keywords: finite element
zbMATH Keywords: condition number
zbMATH Keywords: stability
Wikidata QID: Q57397408
reviewed by: Olof B. Widlund
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 720557