Notes on Numerical Fluid Mechanics

Transcription

1 Notes on Numerical Fluid Mechanics Volume 31 Parallel Algorithm,s for Partial Differential Equations Edited by Wolfgang Hackbusch al V1eweg

2 Parallel Algorithms for Partial Differential Equations Proceedings of the Sixth GAMM-Seminar, Kiel, January 19-21,1990 Edited by Wolfgang Hackbusch vieweg UNIVERSITATSBIBLSOTHEK HANNOVER TECHN1SCHE INFORMATIONSBIBLIOTHEK

3 CIP-Titelaufnahme der Deutschen Bibliothek Parallel algorithms for partial differential equations: Kiel, January 19-21, 1990 / ed. by Wolfgang Hackbusch. - Braunschweig; Wiesbaden: Vieweg, 1991 (Proceedings of the... GAMM seminar; 6) (Notes on numerical fluid mechanics; Vol. 31) ISBN NE: Hackbusch, Wolfgang \Hrsg.} ; Gesellschaft für Angewandte Mathematik und Mechanik: Proceedings of the... ; 2. GT Manuscripts should have well over 100 pages. As they, ill be rep.rodu ed pho ome b.anicall they should be typed with utmost care on special stationary which will be supplied on reque In print, the size will be reduced linearly to approximat ely 75 per cen. F d~ should be lettered accordingly so as to produce letters not smaller than mm prin The is valid for handwritten formulae. Manuscripts (in English} or p roposals hould be n general editor Prof. Dr. E.H. Hirschel, Herzog-Heinrich-Weg 6, D-8011 Zorned Vieweg is a subsidiary company of the Bertelsman n Publi hing Group ln emationa All rights reserved Friedr. Vieweg & Sohn Verlagsgesellschaft mb H, Braun chwei 1991 No part of this publication may be reproduced, ored in a cetrie" system or transmitted, mechanical, photocop in or o ~ ;i ou prior p ermission of the copyright h older. Produced by W. Langelüddecke, Braunschweig Printed in the Federal Republic of Germany ISSN ISBN

4 FOREWORD The GAMM Committee for "Efficient Numerical Methods for Partial Differential Equations" organises workshops on subjects concerning the algorithmical treatment of partial differential equations. The topics are discretisation methods like the finite element and the boundary element method f or various types of applications in structural and fluid mechanics. Particular attention is devoted to the advanced solution techniques like e.g. multi-grid methods. The series of such workshops was continued in 1990, January 19-21, with the 6th Kiel-Seminar on the special topic "Parallel Algorithms for PDEs" at the Christian-Albrechts-University of Kiel. The seminar was attended by 90 scientists from 10 countries and 22 lectures were given. In many lectures applications of the domain decomposition are considered. Combinations with multi-grid methods, conjugate gradient methods, adaptive refinement, and boundary collocation methods are mentioned. But also the orginal domain decomposition algorithm is described. Another decomposition is the problem decomposition, which offers a new kind of parallelism. A parallel variant of the conjugate gradient method is described. Concerning multi-grid methods quite new features are introduced: correction schemes replacing the coarse-grid correction and so-called sparse grids. The latter co~struc~ion leads to completely different discrete problems, wh1ch give accurate results although the dimension of the system is strongly reduced. All topics mentioned above are contained in the 19 contributions to this volume. The editor thanks the Land Schleswig-Holstein for its support. He also likes to express his gratitude to all persons involved in the organisation of the seminar. Kiel, July 1990 W. Hackbusch

5 CONTENTS E. ALLGOWER, K. BOHMER, M. ZHEN: On a problem decomposition for semilinear nearly symmetric elliptic problems 1 P.BASTIAN, J. BURMEISTER, G. HORTON: Implementation of a parallel multi-grid method for parabolic partial differential equations 18 P.E. BJORSTAD, R. MOE, M. SKOGEN: Parallel decomposition and iterative refinement algorithms 28 Th. DREYER: Nonlinear ilu-decomposition as smoother in multigrid methods 47 M. DRYJA, O.B. WIDLUND: Multilevel additive methods for elliptic finite element problems 58 R.E. EWING, R.D.LAZAROV, P.S. VASSILEVSKI: Finite difference schemes on grids with local refinement in time and in space for parabolic problems. Optimal order two-grid iterative methods 70 M. GRIEBEL: A parallelizable and vectorizable multi-level algorithm on sparse grids 94 G. HAASE, U. LANGER: On the use of multigrid preconditioners in the domain decomposition method 101 W. HACKBUSCH: A parallel variant of the conjugate gradient method Ill G. HORTON: Parallel solution of the Navier-Stokes equations in complex geometries 120 B. KISS, G. MOLNARKA: A preconditioned domain decomposition algorithm for the solution of the elliptic Neumann problem. 129 J.A. KOLODZIEJ, G. MUSIELAK: Domain decomposition in boundary collocation method 140 A. KRECHEL, H.-J. PLUM: A parallel block preconditioned conjugate gradient method 157 R.W. LELAND, J.S. ROLLETT: A comparison of parallel extrapolation methods 177 V. MEHRMANN: Divide and conquer methods for tridiagonal linear systems 188

6 H. SBOSNY, K. WITSCH: Parallel multigrid using domain decomposition St. VANDEWALLE, R. PIESSENS: A parallel and vectorizable algorithm for solving parabolic partial differential equations G. WITTUM: An ILU-based smoothing correction scheme Ch. ZENGER: Sparse Grids 241 List of participants 252