A FETI method with a mesh independent condition number for the iteration matrix
type: Item
instance of: scholarly article
title: A FETI method with a mesh independent condition number for the iteration matrix
zbMATH Open document ID: 1186.65147
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 2009-04-23T00:00:00Z
review text: The authors introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of \(H_{0}^{1}(\Omega)\) derived by \textit{P. A. Raviart} and \textit{J. M. Thomas} [Math. Comput. 31, 391--413 (1977; Zbl 0364.65082)] and complemented with the detailed work on polygonal domains developed by \textit{P. Grisvard} [Recherches en Mathématiques Appliquées. 22. Paris: Masson. Berlin: Springer-Verlag. (1992; Zbl 0766.35001)]. They consider the following model problem \[ \begin{aligned} -\Delta u + u & = f \quad \text{in }\Omega,\\ u & = 0 \quad \text{on }\partial\Omega. \end{aligned} \] The authors compute the action of the Lagrange multipliers using the natural \(H_{00}^{1/2}\) scalar product, therefore no consistency error appears. Their analysis allows to deal with cross points and floating subdomains in a natural manner. As a byproduct, they obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number given by \((1 + \log (H/h))^{2}\), where \(H\) and \(h\) are the characteristic subdomain size and element size respectively, shown by \textit{J. Mandel} and \textit{R. Tezaur} [Numer. Math. 73, No.~4, 473--487 (1996; Zbl 0880.65087)]. Finally, numerical results that confirm the presented theoretical analysis are available.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 5546679
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: finite element tearing and interconnecting (FETI) methods
zbMATH Keywords: consistency
zbMATH Keywords: condition number
zbMATH Keywords: numerical results
author: Christine Bernardi
author: Tomás Chacón Rebollo
author: E. Chacón Vera
reviewed by: Petr Necesal
MaRDI profile type: Publication
full work available at URL: j.cma.2007.11.019
OpenAlex ID: W2078599880
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DOI: 10.1016/J.CMA.2007.11.019
description: scientific article; zbMATH DE number 5546679