Fitted numerical methods for singular perturbation problems : error estimates in the maximum norm for linear problems in one and two dimensions PDF
Preview Fitted numerical methods for singular perturbation problems : error estimates in the maximum norm for linear problems in one and two dimensions
Fitted Numerical Methods for Singular Perturbation Problems Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions Revised Edition 8410.9789814390736-tp.indd 1 1/19/12 9:10 AM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Fitted Numerical Methods for Singular Perturbation Problems Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions Revised Edition j j h miller Trinity College, Dublin, Ireland E O’riordan Dublin City University, Ireland G I Shishkin Russian Academy of Sciences, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8410.9789814390736-tp.indd 2 1/19/12 9:10 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FITTED NUMERICAL METHODS FOR SINGULAR PERTURBATION PROBLEMS Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (Revised Edition) Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4390-73-6 ISBN-10 981-4390-73-9 Printed in Singapore. HeYue - Fitted Numerical Methods.pmd 1 1/17/2012, 5:32 PM January4,2012 14:48 WorldScientificBook-9inx6in MOS-ws-book9x6 To Kathy, Lida and Mary v TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January4,2012 14:48 WorldScientificBook-9inx6in MOS-ws-book9x6 Preface This is a monograph about numerical methods for solving singularly per- turbed differential equations. It is a theoretical book, but, throughout the text,thereaderisreferredtopaperspublishedintheliterature,whichcon- tain material about the implementation of these methods and the results of extensive computations on test problems. These numerical methods are simple to describe and easy to implement. It is the proof of the theoretical results that is difficult. The topic of the present book is the theory for lin- ear problems in one dimension, and in two dimensions when the solutions have only regular layers. Since most of the ideas and techniques presented here have not previously been published in detail in book form, its goal is toexplaintheminareasonablysimpleway. Therefore,noattemptismade tobecomprehensive, nortostateandprovetheresultsinthemostgeneral case. Instead the key ideas are explained for simple problems containing the crucial difficulties. The most general cases require more complicated constructions, and proofs, but do not require further fundamental ideas. Many unsolved problems remain in this interesting and exciting area, but it will be apparent to the reader that much progress has been made since the first monograph [Doolan et al. (1980)] was published. This book falls naturally into three parts. The first three chapters providemotivation,andanelementaryintroduction,tosomeaspectsofthe subject. The next seven chapters are concerned with problems exclusively in one dimension, while, in the final five chapters, problems in two spatial dimensions, or in one spatial dimension and time, are considered. In the first three chapters simple examples of various one-dimensional problems involving singular perturbations are described, and some issues concerningtheirnumericalsolutionarediscussed. Thefactthatsuchprob- lemscannotbesolvednumerically, inacompletelysatisfactorymanner, by vii January4,2012 14:48 WorldScientificBook-9inx6in MOS-ws-book9x6 viii Fitted Numerical Methods for SPPs standard numerical methods, is then explained. This indicates the need for methods that behave uniformly well, whatever the value of the singu- lar perturbation parameter. Such methods are called ε-uniform methods, where ε is the singular perturbation parameter. Simple examples of ε-uniform finite difference methods are present in Chapters 4 and 5. These are of two kinds: the first are the fitted operator methods, which comprise specially designed finite difference operators on standard meshes; the second are the fitted mesh methods, which comprise standard finite difference operators on specially designed meshes. In Chapter 4 fitted operator methods on uniform meshes are described for some simple problems in one dimension. The chapter concludes with the construction of the El-Mistikawy Werle fitted operator method, for linear convection-diffusion equations in one dimension, and with a modern proofthatitisanε-uniformmethod. InChapter5fittedmeshmethodsfor simpleproblemsinonedimensionareconstructed. Asimplebasiclemmais established and it is then proved that fitted mesh methods for initial value problemsareε-uniform. InChapter6itisprovedthatfittedmeshmethods, forlinearreaction-diffusionequationsinonedimension,areε-uniform. The next two chapters are concerned with linear convection-diffusion problems in one dimension. Chapter 7 contains technical results about upwind finite difference operators on fitted meshes, which are required for the proof in Chapter8thatsuchfittedmeshmethodsareε-uniformfortheseproblems. InChapter9finiteelementmethodsonfittedmeshes,forlinearconvection- diffusionproblems inonedimension, areconstructedanda proof that they areε-uniformisgiven. TheuseoftheSchwarziterativemethodisillustrated in Chapter 10, where it is applied to the one dimensional linear reaction- diffusion equation. A proof that the method is ε-uniform is also presented. The remainder of the book is devoted to problems in two dimensions. Several linear convection-diffusion problems in two dimensions, and their numerical solution, are described in Chapter 11. In Chapter 12 bounds are obtained for derivatives of the solutions of such problems, in the case where only regular layers occur. Then, in Chapter 13, these bounds are usedtoestablishthefactthatthefittedmeshmethod,constructedforthese problems in Chapter 11, is ε-uniform. Chapter 14 contains the surprising result that it is impossible to construct an ε-uniform numerical method, usingafittedoperatormethodonuniformrectangularmeshes,forproblems withparabolicboundarylayers. Itisalsoindicatedthat,forsuchproblems, ε-uniform fitted mesh methods are quite easy to construct. Finally, in Chapter 15, it is proved that it is impossible to construct an ε-uniform January4,2012 14:48 WorldScientificBook-9inx6in MOS-ws-book9x6 Preface ix numerical method, using a standard finite difference operator on a fitted rectangular mesh, for a problem having both an initial and a parabolic boundary layer. It is also indicated that, for such problems, ε-uniform numerical methods can be constructed, using both a fitted operator and a fitted mesh. The book ends with an appendix, which contains a brief reviewofsomeclassicalboundsonthederivativesofthesolutionsofpartial differential equations, stated in the terminology used in this book. One of the main messages in this book is the great importance of using the appropriate mesh to solve a problem numerically. The choice of an appropriatemeshhasreceivedlessdetailedattentionintheliterature,than theconstructionofanappropriatefinitedifferenceoperatororfiniteelement subspace. Certainly, fewer rigorous analytic results have been obtained in this direction. While the great importance of a fitted mesh, for solving singular perturbation problems, is established rigorously in this book, the lastchapterofthebookshowsthatfittedoperatorsareinnosenseobsolete. Indeed, it is proved there, rigorously, that a fitted rectangular mesh is not always sufficient to guarantee that the resulting method is ε-uniform. It follows, therefore, that, for some problems, not only fitted meshes but also fitted operators should be considered. The ε-uniform error estimates in this book are obtained using the fol- lowing approach. The ε-uniform stability of the finite difference operator is established using a maximum principle, whenever this is available. This is the case for all of the numerical methods considered in this book with the exception of the finite element method in Chapter 9. The exact and the discrete solutions of the problem are then decomposed, into smooth and singular components, and the errors in each component are estimated separately. The key step, needed to obtain these estimates, is the estab- lishment of suitable bounds on the derivatives of the smooth and singular components of the solution. A number of key points should be emphasised. First, the fitted numer- ical methods discussed in this book are designed to be robust with respect to changes in the singular perturbation parameter. Secondly, the error es- timates, obtained here, are valid at each point of the mesh or domain, and they are measured in the maximum norm. The choice of the fitting factor, orthe constructionofthe fittedmesh, requires a priori informationabout the location and width of the layers that are to be resolved. Fortunately, such information is frequently available from the mathematical literature ontheasymptoticanalysisofsingularperturbationproblems. Anyscientist or engineer requiring accurate and robust numerical approximations to the
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