Efficient Solutions of Elliptic Systems - Springer Link

Wolfgang Hackbusch

Efficient Solutions of Elliptic Systems

Notes on Numerical Fluid Mechanics Volume 10 Series Editors: Ernst Heinrich Hirschel, MUnchen Maurizio Pandolfi, Torino Arthur Rizzi, Stockholm Bernard Raux, Marseille Volume 1

Boundary Algorithms for Multidimensional lnviscid Hyperbolic Flows (Karl Forster, Ed.)

Volume 2

Proceedings of the Third GAMM·Conference on Numerical Methods in Fluid Mechanics (Ernst Heinrich Hirschel, Ed.) (out of print)

Volume 3

Numerical Methods for the Computation of lnviscid Transonic Flows with Shock Waves (Arthur Rizzi/Henri Viviand, Eds.)

Volume 4

Shear Flow in Surface-Oriented Coordinates (Ernst Heinrich Hirschei/Wilhelm Kordulla)

Volume 5

Proceedings of the Fourth GAMM-Conference on Numerical Methods in Fluid Mechanics (Henri Viviand, Ed.) (out of print)

Volume 6

Numerical Methods in Laminar Flame Propagation (Norbert Peters/Jiirgen Warnatz, Eds.)

Volume 7

Proceedings of the Fifth GAMM-Conference on Numerical Methods in Fluid Mechanics (Maurizio Pandolfi/Renzo Piva, Eds.)

Volume 8

Vectorization of Computer Programs with Applications to Computational Fluid Dynamics (Wolfgang Gentzsch)

Volume 9

Analysis of Laminar Flow over a Backward Facing Step (Ken Morgan, Jaques Periaux, Fran~ois Thomasset, Eds.)

Volume 10 Efficient Solutions of Elliptic Systems (Wolfgang Hackbusch, Ed.) Manuscripts should have well over 100 pages. As they will be reproduced toto· mechanically they should be typed with utmost care on special stationary which will be supplied on request. In print, the size will be reduced linearly to approxi· mately 75 %. Figures and diagrams should be lettered accordingly so as to produce letters not smaller than 2 mm in print. The same is valid for handwritten formulae. Manuscripts (in English) or proposals should be sent to the general editor Prof. Dr. E. H. Hirschel, MBB-LKE 122, Postfach 80 11 60, D-8000 Munchen 80.

Wolfgang Hackbusch (Ed.)

Efficient Solutions of Elliptic Systems Proceedings of a GAMM-Seminar Kiel, January 27 to 29, 1984

Springer Fachmedien Wiesbaden GmbH

CIP-Kurztitelaufnahme der Deutschen Bibliothek

Efficient solutions of elliptic systems: proceedings of a GAMM seminar Kiel, January 27 to 29, 1984 1 Wolfgang Hackbusch (ed.). (Notes on numerica! fluid mechanics; Voi. 10) ISBN 978-3-528-08084-6 ISBN 978-3-663-14169-3 (eBook) DOI 10.1007/978-3-663-14169-3 NE: Hackbusch, Wolfgang [Hrsg.); Gesellschaft fiir Angewandte Mathematik und Mechanik; GT

Ali righ ts reserved © Springer Fachmedien Wiesbaden 1984 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig in 1984

No part of this publication may be reproduced, stored in a retrieval system or transmitted mechanically, by photocopies, recordings or otherwise, without prior permission of the copyright holder. Produced by Industrie u. Verlagsdruck, Walluf b. Wiesbaden

ISBN 978-3-528-08084-6

FO:REWORD The GAMM-Committee for Efficient Numerical Methods for PDE (GAMM-FachausschuB "Effiziente numerische Verfahren flir oartielle Differentialgleichungen") organises conferences anc seminars on subjects concerning the algorithmic treatment of pde problems. The topic "EFFICIENT SOLUTION OF ELLIPTIC SYSTE~S" of a G~.M­ Seminar held at the University of Kiel, January 27-29, 1984, plays a central role in structural and fluid mechanics. For instance, such elliptic systems are formed by the equations of Stokes and Navier-Stokes. The discretisation eq by finite elements and the iterative solution of the arising discrete equations are more difficult than for single elliptic equations. The Seminar was attended by 43 scientists from 12 countries. It was possible to bring together experts in particular from the fields of finite element methods and multi-grid methods. Accordingly most of the 17 papers presented at the Seminar concerned tne finite element discretisation and the multi-grid solution of the discretised problems. These proceedings contain 11 contributions in alphabetical order. The editor, who has also been the organiser of this Seminar, would like to acknowledge a support from the land SchleswigHolstein.

Kiel, April 1984

w.

Hackbusch

CONTENTS Page P. BJ~RSTAD: The Direct Solution of a Generalised Biharmonic Equation on a Disk ....•............•......... F. BREZZI, J. PITKKRANTA: On the Stabilization of Finite Element Approximations of the Stokes Equations ......... . E. DICK: A Multigrid H.ethod for Cauchy-Riemann and Steady Euler Equations Based on Flux-Difference Splitting .•••.• L. FUCHS: Multi-Grid Schemes for Incompressible Flows ..• L. FUCHS: Defect Corrections and Higher Numerical Accuracy .•......•....••.•••.................•..•.......• F.K. HEBEKER: On a Multigrid Method to Solve the Integral Equations of 3-D Stokes' Flow .•..•..•..•................ u\." V /V • P. NEITTAANr~KI, M. KRIZEK: Conforming FE-Method for Obtaining the Gradient of a Solution to the Poisson Equation ••.•.••••••.••.•....•...•.•.•.....•.......•..•.. B. RUTT~~NN, K. SOLCHENBACH: A ~ultigrid Solver for the In-cylinder Turbulent Flows in En9ines ......•...•......

11

20 38

52 67

74

87

U. SCHUMANN, H. VOLKERT: Three-Dimensional Mass- and 109 Helmholtz-Equation in TerrainFollowing Coordinates ••..........••..•.........•....•..• R. VERFURTH: Numerical Solution of Mixed Finite Element 132 Problems .••..••••.•......•......................•......• 145 P. WESSELING: Multigrid Solution of the Navier-Stokes Equations in the Vorticity-Streamfunction Formulation ... Momentu~-Consistent