Multigrid Methods IV: Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6–9, 1993

359 Pages·1994·7.75 MB·English

by  P. Bastian

ISNM
International Series of
Numerical Mathematics
Vol. 116
Edited by
K.-H. Hoffmann, München
H. D. Mittelmann, Tempe
J. Todd, Pasadena
Multigrid Methods
Proceedings of the Fourth European Multigrid Conference,
Amsterdam, July 6-9,1993
Edited by
P.W. Hemker
P. Wesseling
Springer Basel AG
Editors
P.W. Hemker P. Wesseling
CWI Delft University of Technology
Dept. of Numerical Mathematics Faculty of Techn. Mathematics & Informatics
Kruislaan 413 Mekel weg 4
1098 SJ Amsterdam 2628 CD Delft
The Netherlands The Netherlands
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Multigrid methods IV: proceedings of the Fourth European
Multigrid Conference, Amsterdam, July 6-9,1993. Ed. by P.
W. Hemker; P. Wesseling. - Basel; Boston ; Berlin :
Birkhäuser, 1994
(International series of numerical mathematics ; Vol 116)
ISBN 978-3-0348-9664-1 ISBN 978-3-0348-8524-9 (eBook)
DOI 10.1007/978-3-0348-8524-9
NE: Hemker, Pieter W. [Hrsg.]; European Multigrid Conference <4,
1993, Amsterdam^ GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on
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be obtained,
© 1994 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel, Switzerland in 1994
Softcover reprint of the hardcover 1st edition 1994
Camera-ready copy prepared by the editors
Printed on acid-free paper produced from chlorine-free pulp
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9664-1
987654321
Contents
I Invited Papers
1 On Robust and Adaptive Multi-Grid Methods 1
P. Bastian and G. Wittum
2 A Generalized Multigrid Theory in the Style of Standard Iterative
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
Craig C. Douglas
3 Turbulence Modelling as a Multi-Level Approach 35
L. Fuchs
4 The Frequency Decomposition Multi-Grid Method. . . . . . . . .. 43
Wolfgang H ackbusch
5 Multiscale Methods for Computing Propagators in Lattice Gauge
Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57
P. G. Lauwers
6 Adaptive Multigrid on Distributed Memory Computers . . . . . .. 77
Hubert Ritzdorf and Klaus Stuben
7 Multicomputer-Multigrid Solution of Parabolic Partial Differential
Equations ................................ 97
Stefan Vandewalle and Graham Horton
8 Multilevel Solution of Integral and Integro-differential Equations in
Contact Mechanics and Lubrication. . . . . . . . . . . . . . . . .. 111
C.H. Venner and A.A. Lubrecht
Contributed Papers
1 A Multi-Grid Method for Calculation of Turbulence and Combustion 131
X.S. Bai and L. Fuchs
2 On a Multi-Grid Algorithm for the TBA Equations . . . . . . . .. 143
Alfio Borzi and Anni Koubek
3 A Multidimensional Upwind Solution Adaptive Multigrid Solver for
Inviscid Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . .. 151
L. A. Catalano, P. De Palma, M. Napolitano and G. Pascazio
4 Parallel Steady Euler Calculations using Multigrid Methods and
Adaptive Irregular Meshes . . . . . . . . . . . . . . . . . . . . . .. 163
VI
J. De Keyser and D. Roose
5 Multigrid Methods for Steady Euler Equations Based on Multi-stage
Jacobi Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175
Erik Dick and K ris Riemslagh
6 Multigrid and Renormalization for Reservoir Simulation ...... 189
Michael G Edwards and Clive F Rogers
7 Interpolation and Related Coarsening Techniques for the Algebrai~
Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
G. Golubovici and C. Popa
8 Parallel Point-oriented Multilevel Methods . . . . . . . . . . . . . . 215
Michael Griebel
9 Large Discretization Step (LDS) Methods For Evolution Equations 233
Zigo Hams and Shlomo Ta 'as an
10 A Full Multigrid Method Applied to Turbulent Flow using the SIM
PLEC Algorithm Together with a Collocated Arrangement . . . . . 245
Peter Johansson and Lars Davidson
11 Multigrid Methods for Mixed Finite Element Discretizations of Vari
ational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Tilman Neunhoeffer
12 Multigrid with Matrix-dependent Transfer Operators for Convection
diffusion Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Arnold Reusken
13 Multilevel, Extrapolation, and Sparse Grid Methods. 281
U. Riide
14 Robust Multi-grid with 7-point ILU Smoothing 295
Rob Stevenson
15 Optimal Multigrid Method for Inviscid Flows . 309
Shlomo Ta'asan
16 Multigrid Techniques for Simple Discretely Divergence-free Finite
Element Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Stefan Turek
17 Grid-independent Convergence Based on Preconditioning Techniques 333
A. van der Ploeg, E.F.F. Botta and F. W. Wubs
18 A New Residual Smoothing Method for Multigrid Acceleration Ap-
plied to the Navier-Stokes Equations . . . . . . . . . . . . . . . . . 345
Zhong Wen Zhu, Chris Lacor and Charles Hirsch
Preface
This volume contains a selection from the papers presented at the Fourth European
Multigrid Conference, held in Amsterdam, July 6-9,1993. There were 78 registered
participants from 14 different countries, and 56 presentations were given.
The preceding conferences in this series were held in Cologne (1981, 1985) and
in Bonn (1990). Also at the other side of the Atlantic special multigrid conferences
are held regularly, at intervals of two years, always in Copper Mountain, Colorado,
US. The Sixth Copper Mountain Conference on Multigrid Methods took place
in April, 1993. Circumstances prevented us from putting a larger time interval
between the Copper and Amsterdam meetings. The next European meeting is
planned in 1996, a year later than the next Copper Meeting.
When the first multigrid conference was held in 1981 there was no doubt
about the usefulness of a conference dedicated specially to multigrid, because
multigrid was a new and relatively unexplored subject, still in a pioneering stage,
and pursued by specialists. The past twenty years have shown a rapid growth in
theoretical understanding, useful applications and widespread acceptance of multi
grid in the applied disciplines. Hence, one might ask whether there is still a need
today for conferences specially dedicated to multigrid. The general consensus is
that the answer is affirmative. New issues have arisen that are best addressed or
need also be addressed from a special multigrid point of view. Most prominent
among these issues are parallel computing, adaptive computations and applica
tions other than elliptic boundary value problems. Multigrid has much impact on
computational fluid dynamics, but also in other fields profitable use of multilevel
concepts is possible and starts to develop. In fact, in almost all areas in which
intensive computing is a major tool, multilevel principles may bring improvements
or even allow major breakthroughs. Hence, to exchange the experiences special
multigrid conferences will continue to be useful in the foreseeable future.
Exchange of information on multigrid research is further aided by MGNet, in
which papers and software are stored electronically, and may be retrieved by ftp.
MGNet is maintained by C. Douglas of Yale University. Information on MGNet
can be obtained by sending email to [email protected].
The papers in this volume are ordered alphabetically by author. The invited
presentations are followed by a selection of contributed papers. Financial con
straints put a page limit on this volume. Rather than severely limit the number
viii
of pages available for each contribution, reducing these more or less to technical
abstracts, we preferred to give authors sufficient space to show interesting details,
and to accept the consequence, that not all contributions could find a place in these
pages. We made a selection, and the remaining contributions will be published by
the Centre for Mathematics and Informatics (CWI) in Amsterdam in their CWI
Tract series.
Several trends in the field that are discernible at present, are reflected in the
papers presented at the conference. Maturing parallel computing technology has an
increasing impact on scientific computing. This development is of prime concern
to multigrid practitioners. After all, reduction of computing cost, measured in
various norms, such as financial cost or elapsed wall clock time has from the
start been the most (though not the only) appealing aspect of multigrid from a
practical point of view. The holy grail of "just a few work units" takes on a new
aspect in a parallel computing environment. Similar considerations of cost and
quality lead to adaptive discretisation techniques for those applications that go
beyond the realm of the smooth and continuous, to include sharp-edged features
and discontinuities. Multigrid is especially suited here, because of the possibilities
it offers for a-posteriori error estimation, and hence for the detection of special
structures in solutions. Furthermore, multigrid for unstructured grids is actively
pursued. In three-dimensional domains of complicated shape, unstructured grids
are much more easily generated than structured grids and they give more flexibility
when adapting the grid to the behaviour of the solution. However, there is still
a long way to go before efficiency similar to that obtained for structured grids is
obtained, especially for equations of second order. Algebraic multigrid, in which
no reference is made at all to an underlying grid structure, shows progress, but
also needs to be developed further.
Multigrid has become an indispensible tool in computational fluid dynamics.
Significant new developments are seen in the treatment of evolution and hyper
bolic problems. Steady progress is being made for the multifarious mathematical
models that play a role in fluid dynamics. But other fields present huge compu
tational challenges as well. A prime example is quantum chromodynamics. Aided
by multigrid, significant breakthroughs seem in the offing. Another example is tri
bology, where computational models have improved significantly by application of
multigrid. The reader will find papers about these topics in the present volume.
The conference was made possible by the Centre of Mathematics and Infor
matics (CWI), Amsterdam, and the University of Amsterdam. Financial support
was provided by Akzo NV, IBM Nederland NV and the Royal Dutch Academy
of Science (KNAW). We are also greatly indebted to Mr Frans Snijders and Ms
Simone van der Wolff for their help in organising the conference in the historic
setting of old Amsterdam.
Amsterdam / Delft, October 1993
P.W. Hemker
P. Wesseling
Part I
Invited Papers
1
On Robust and Adaptive
Multi-Grid Methods
P. Bastian and G. Wittum1
ABSTRACT In the present paper we discuss the development and practical ap
plication of robust multi-grid methods to solve partial differential equations on
adaptively refined grids. We review several approaches to achieve robust multi-grid
methods and describe two special new strategies for anisotropic and convection
diffusion problems. The performance of these algorithms is investigated for three
selected test problems.
1 Introduction
In the present paper we discuss the development and practical application of ro
bust multi-grid methods to solve partial differential equations on adaptively re
fined grids. Since a couple of years multi-grid methods are well established as fast
solvers for large systems of equations arising from the discretization of differen
tial equations. However, it is still a substantial unresolved question to find robust
methods, working efficiently for large ranges of parameters e.g. in singularly per
turbed problems. This applies to diffusion-convection-reaction equations, arising
e.g. from modelling of flow through porous media, the basic equations of fluid
mechanics and plate and shell problems from structural mechanics.
Multi-grid methods are known to be of optimal efficiency, i.e. the convergence
rate K does not depend on the dimension of the system, characterized by a stepsize
h. Following [28] we call a multi-grid method robust for a singularly perturbed
problem, if
K(h,e) ::; KO < 1, \Ie> 0, h > 0, (1)
e denoting the singular perturbation parameter. Up to now multi-grid methods
satisfying (1) have been studied in the literature only for special model cases using
structured grids, see [25], [26], [15], [27], [28], [29].
llnterdiszipliniires Zentrum fiir Wissenschaftliches Rechnen (IWR), Universitiit Hei
delberg, 1m Neuenheimer Feld 368, 69120 Heidelberg, Federal Republic of Germany,
email: wittum(Diwr.uni-heidelberg.de