Lectures on The Finite Element Method By Ph. Ciarlet TataInstituteofFundamentalResearch Bombay 1975 Lectures on The Finite Element Method By Ph. Ciarlet Notesby S. Kesavan,Akhil Ranjan M. Vanninathan TataInstituteofFundamentalResearch Bombay 1975 (cid:13)c TataInstituteofFundamentalResearch,1975 No part of this book may be reproduced in any formbyprint, microfilmoranyothermeanswith- out written permission from the Tata Institute of Fundamental Research, Colaba,Bombay400005 PrintedInIndia By AnilD.VedAtPrabhatPrinters, Bombay400004 AndPublished By TheTataInstituteofFundamentalResearch, Bombay. Acknowledgements THE AUTHOREXPRESSEShis deep gratitude to Professor Ramana- than for his very kind invitation to deliver these lectures at Bangalore, aspartoftheT.I.F.R.–I.I.Sc. MathematicsProgramme. Also,theauthor tankstheT.I.F.R.forfinancialsupport. Ph. Ciarlet v Preface OUR BASIC AIM has been to present some of the mathematical as- pects of the finite element method, as well as some applications of the finite element method for solving problems in Elasticity. This is why important topics, such as curved boundaries, mixed and hybrid meth- ods, time-dependent problems, etc..., are not covered here. No attempt hasbeenmadetogiveanexhaustive bibliography. vii Contents Acknowledgements v Preface vii 1 TheAbstractProblem 1 2 Examples 9 3 TheFiniteElementMethodinitsSimplestForm 29 4 ExamplesofFiniteElements 35 5 GeneralPropertiesofFiniteElements 53 6 Interpolation TheoryinSobolevSpaces 59 7 ApplicationstoSecond-OrderProblems... 67 8 NumericalIntegration 77 9 TheObstacleProblem 95 10 ConformingFiniteElementMethodforthePlateProblem 103 11 Non-ConformingMethodsforthePlateProblem 113 ix