Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations


by

Owe Axelsson, Janos Karatson

DOI: 10.2174/97816080529121110101
eISBN: 978-1-60805-291-2, 2011
ISBN: 978-1-60805-610-1



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Indexed in: Scopus

This e-book presents several research areas of elliptical problems solved by differential equations. The mathematical models explaine...[view complete introduction]

Table of Contents

Foreword

- Pp. i-ii (2)

Radim Blaheta

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Preface

- Pp. iii

Owe Axelsson and Janos Karatson

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List of Contributors

- Pp. iv-v (1)

Owe Axelsson and Janos Karatson

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Robust Multilevel Preconditioning Methods

- Pp. 3-22 (20)

Petia Boyanova and Svetozar Margenov

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Efficient Preconditioners for Saddle Point Systems

- Pp. 23-43 (21)

Zhi-Hao Cao

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Block Preconditioners for Saddle Point Problems Resulting from Discretizations of Partial Differential Equations

- Pp. 44-65 (22)

Piotr Krzyzanowski

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Construction of Preconditioners by Mapping Properties for Systems of Partial Differential Equations

- Pp. 66-83 (18)

Kent-Andre Mardal and Ragnar Winther

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Automatic Construction of Sparse Preconditioners for High-Order Finite Element Methods

- Pp. 84-102 (19)

Travis M. Austin, Marian Brezina, Thomas A. Manteuffel and John Ruge

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A Geometric Toolbox for Tetrahedral Finite Element Partitions

- Pp. 103-122 (20)

Jan Brandts, Sergey Korotov and Michal Krizek

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Local Multigrid on Adaptively Refined Meshes and Multilevel Preconditioning with Applications

- Pp. 123-144 (22)

Ronald H.W. Hoppe, Xuejun Xu and Huangxin Chen

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Index

- Pp. 145-146 (2)

Owe Axelsson and Janos Karatson

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Foreword

The present book is a collection of chapters written by outstanding specialists, and is devoted to one of the most challenging topics of contemporary numerical mathematics. The topic – development of efficient preconditioners and solution methods for (discretized) elliptic partial differential equations – is crucial for the mathematical modelling of complex physical and other processes in most branches of science and engineering.

Efficient solution methods usually arise as a combination of a suitable iterative technique and an efficient preconditioner, which is typically problem specific, using a proper approximation to the solved problem with further attributes as cheap actions with it, a possibility of parallel implementation, numerical and parallel scalability and robustness with respect to the problem parameters. The development of preconditioners has now its own history starting from matrix splittings, diagonal scaling and incomplete factorization for model elliptic problems and continuing with multigrid and multilevel methods, approximate inverses, domain decomposition methods and other approaches to the solution of complicated elliptic problems discretized by the finite element or other discretization methods. A substantial progress in the field of preconditioners has been achieved, but at the same time many new questions and challenges have arisen, and this is why the research is even more intensive nowadays than it was previously.

The present book covers many topics of the contemporary research in the field of efficient preconditioners, namely:

Chapter 1, written by P. Boyanova and S. Margenov, concerns the Algebraic Multilevel Iterative (AMLI) methods first introduced by Axelsson and Vassilevski. The paper overviews the methods with a special care of efficiency and robustness as well as application to both elliptic and parabolic problems discretized by either conforming or nonconforming finite element methods.

Chapter 2 by Z.-H. Cao concerns efficient preconditioning of saddle point systems, which is the timely topic appearing also in further chapters. The paper examines several preconditioners arising from either factorization or augmentation of saddle point systems with a careful examination of spectra and structure of the preconditioned matrices.

Also Chapter 3 by P. Krzyzanowski considers preconditioners for saddle point systems with an emphasis on construction of efficient block preconditioners scalable with respect to the mesh size.

Chapter 4 by K.-A. Mardal and R.Winther discusses a general approach to the construction of preconditioners, which explains scalable properties of preconditioners and can be used for the construction of new ones. It also provides examples illustrating the theory.

Chapter 5 by T. M. Austin, M. Brezina, T. A. Manteuffel and J. Ruge is devoted to preconditioning of systems arising from high order (p-version) FEM. A new procedure is described and tested, which constructs a sparse approximation to the FEM matrix by solving local eigenvalue and least-squares problems and then define the preconditioner by AMG approximation of the inverse of this sparse matrix.

Chapter 6 by J. Brandts, S. Korotov and M. Křížek is an exception as it is not directly devoted to the preconditioners but to tetrahedral finite element partitions. Nevertheless, these results are important for the analysis of the finite element systems, mesh refinements and also e.g. AMLI type preconditioners depending on the angle between a coarse grid space and its hierarchic completion.

Finally, Chapter 7 by R.H.W. Hoppe, X. Xu and H. Chen considers local multigrid methods applied to complicated problems from electrical engineering and acoustic.

Generally, the book provides a lot of information on recent achievements in several fields of the development of efficient preconditioners. The book appears also thanks to the publisher’s understanding of the importance of such a volume and to the effort of Owe Axelsson and János Karátson, who made a selection and careful edition of the contributions. I hope that the readers appreciate this book providing an excellent overview of many new results and representing a successful joint work of authors, editors, reviewers and the publisher.

Radim Blaheta

7 November 2010

Institute of Geonics, Academy of Sciences of the Czech Republic

Studentska 1768, 708 00, Ostrava-Poruba, Czech Republic


Preface

The goal of this book is to highlight some central areas in current research on the numerical solution of elliptic problems. Mathematical models involving elliptic partial differential equations arise in a variety of real-life problems in science and engineering. Besides phenomena fully described by an elliptic equation, also time-dependent problems describing various evolutionary processes often lead to elliptic problems as subproblems arising in the course of the solution procedure. Furthermore, saddle-point problems of Stokes type, which are not elliptic in a strict sense, can be considered elliptic in a wider sense, being stationary problems that can be reduced to coercivity via the Schur complement operator. These facts reinforce the fundamental role of elliptic problems and their efficient numerical solution.

Solution methods for elliptic problems have undergone great development, and have resulted in many cases in efficient optimal algorithms, on the other hand, various new challenges induce much further research. The discretization of elliptic problems leads to algebraic systems often of very large size. To save computer memory and elapsed time, such problems are normally solved by iteration, most commonly using a preconditioned conjugate gradient (PCG) method. The proper preconditioning technique is a crucial part of the efficient solution of the arising linear systems, and this forms a major topic of this book. Particular attention is paid to multigrid and multilevel methods, and to preconditioning for saddle-point problems, often in block form. Higher order discretization methods and finite element mesh generation are also considered.

Altogether, this book provides a careful presentation of major fields in solving elliptic problems. This is done with the help of leading experts in this topic, who survey the current stage of research in individual chapters. We are convinced that both researchers in the field and users of iterative solution methods for real-life applications will benefit by these valuable contributions. We are grateful for the interest shown in this issue by the authors of these chapters. We thank also our colleague, Radim Blaheta, who due to lack of time was unable to contribute a chapter to this issue, but who kindly offered instead to write the excellent foreword to it.

Owe Axelsson

Institute of Geonics

Academy of Sciences of the Czech Republic.

János Karátson

Institute of Mathematics

ELTE University, Budapest, Hungary.

List of Contributors

Editor(s):
Owe Axelsson
Institute of Geonics
Academy of Sciences of the Czech Republic
Czech Republic


Janos Karatson
Institute of Mathematics
ELTE University
Hungary




Contributor(s):
Austin Travis
Tech-X Corporation
5621 Arapahoe Ave
5621 Arapahoe Ave
CO 80303



Boyanova Petia
Institute of Information and Communication Technologies
Bulgarian Academy of Sciences Acad. G. Bonchev str, bl. 25A
1113 Sofia
Bulgaria
/
Department of Information Technology
Uppsala University Box 337
Uppsala University Box 337
Sweden


Brandts Jan
Korteweg-de Vries Institute
University of Amsterdam
Science Park 904
1098 XH Amsterdam
Netherlands


Brezina Marian
Department of Applied Mathematics
University of Colorado 526 UCB Boulder
CO 80309-0526
USA


Cao Zhi-Hao
School of Mathematical Sciences
and Laboratory of Mathematics for Nonlinear Sciences Fudan University
Shanghai , 200433
People's Republic of China


Chen Huangxin
LSEC, Institute of Computational Mathematics
Chinese Academy of Sciences P.O.Box 2719
Beijing , 100080
People's Republic of China


Hoppe Ronald
Department of Mathematics
University of Houston
Houston
TX , 77204-3008
USA
/
Institute of Mathematics
University at Augsburg
D-86159 Augsburg
Germany


Korotov Sergey
BCAM - Basque Center for Applied Mathematics
Bizkaia Technology Park
Building 500, E-48160 Derio
Basque Country
Spain


Křížek Michal
Institute of Mathematics
Academy of Sciences
Žitná 25, CZ-115 67 Prague 1
Czech Republic


Krzyżanowski Piotr
Institute of Applied Mathematics
University of Warsaw Banacha 2, 02-097
Warszawa
Poland


Manteuffel Thomas A.
Department of Applied Mathematics
University of Colorado 526 UCB Boulder
CO , 80309-0526
USA


Mardal Kent-Andre
Center for Biomedical Computing
Simula Research Laboratory P.O. Box 134, 1325
Lysaker
Norway


Margenov Svetozar
Institute of Information and Communication Technologies
Bulgarian Academy of Sciences
Acad. G. Bonchev str, bl. 25A, 1113
Sofia
Bulgaria


Ruge John
Department of Applied Mathematics
University of Colorado 526 UCB Boulder
CO , 80309-0526
USA


Xu Xuejun
LSEC, Institute of Computational Mathematics
Chinese Academy of Sciences P.O.Box 2719
Beijing , 100080
People's Republic of China


Winther Ragnar
Centre of Mathematics for Applications and Department of Informatics
University of Oslo
0316 Oslo
Norway




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