Excerpt from Studies in Domain Decomposition: Multilevel Methods and the Biharmonic Dirichlet Problem
Multilevel methods, such as multigrid methods, are among the most efficient methods for linear equations arising from elliptic problems; cf. Hackbusch mccormick [38] and the references therein. Recently, with the increasing interest in parallel computation, several new multilevel methods have been developed; cf. Yserentant Bank, Dupont and Yserentant Bramble, Pasciak and Xu and Dryja and Widlund In this thesis, we give improved results for a class of multilevel methods by showing that the condition number of the iteration Operator grows at most linearly with the number of levels in general, and is bounded by a constant independent of the mesh sizes and the number of levels if the elliptic problem is Hz - regular. This is an improvement on Dryja and Widlund's results on a multilevel additive Schwarz method as well as Bramble, P-asciak and Ku's results on the bpx algorithm.
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Excerpt from Studies in Domain Decomposition: Multilevel Methods and the Biharmonic Dirichlet Problem
Multilevel methods, such as multigrid methods, are among the most efficient methods for linear equations arising from elliptic problems; cf. Hackbusch mccormick [38] and the references therein. Recently, with the increasing interest in parallel computation, several new multilevel methods have been developed; cf. Yserentant Bank, Dupont and Yserentant Bramble, Pasciak and Xu and Dryja and Widlund In this thesis, we give improved results for a class of multilevel methods by showing that the condition number of the iteration Operator grows at most linearly with the number of levels in general, and is bounded by a constant independent of the mesh sizes and the number of levels if the elliptic problem is Hz - regular. This is an improvement on Dryja and Widlund's results on a multilevel additive Schwarz method as well as Bramble, P-asciak and Ku's results on the bpx algorithm.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Excerpt from Studies in Domain Decomposition, Vol. 255: Multilevel Methods and the Biharmonic Dirichlet Problem
A class of multilevel methods for second order problems is considered in the additive Schwarz framework. It is established that, in the general case, the condition number of the iterative operator grows at most linearly with the number of levels. The bound is independent of the mesh sizes and the number of levels under a regularity assumption. This is an improvement of a result by Dryja and Widlund on a multilevel additive Schwarz algorithm, and the theory given by Bramble, Pasciak and Xu for the BPX algorithm.
Additive Schwarz and iterative substructuring algorithms for the biharmonic equation are also considered. These are domain decomposition methods which have previously been developed extensively for second order elliptic problems by Bramble, Pasciak and Schatz, Dryja and Widlund and others.
Optimal convergence properties are established for additive Schwarz algorithms for the biharmonic equation discretized by certain conforming finite elements. The number of iterations for the iterative substructuring methods grows only as the logarithm of the number of degrees of freedom associated with a typical subregion. It is also demonstrateed that it is possible to simplify the basic algorithms. This leads to a decrease of the cost but not of the rate of convergence of the iterative methods. In the analysis, new tools are developed to deal with Hermitian elements. Certain new inequalities for discrete norms for finite element spaces are also used.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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Paperback. Zustand: New. Print on Demand. This book explores multilevel methods for solving large-scale second-order elliptic problems. The author considers a class of multilevel methods in the additive Schwarz framework and establishes that under a regularity assumption, the condition number of the iterative operator grows at most linearly with the number of levels. The author extends this idea to the biharmonic Dirichlet problem, and constructs additive Schwarz algorithms for the biharmonic problem using various conforming finite element discretizations. The author establishes optimality and almost optimal convergence properties for the algorithms. The book is a valuable resource for researchers and graduate students in numerical analysis and scientific computing, especially those interested in domain decomposition methods for solving large-scale partial differential equations. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Bestandsnummer des Verkäufers 9781332201211_0
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