The Finite Element Method for Elliptic Problems

by P.G. Ciarlet

Overview

The objective of this book is to analyze within reasonable limits (it is not a treatise) the basic mathematical aspects of the finite element method. The book should also serve as an introduction to current research on this subject.

On the one hand, it is also intended to be a working textbook for advanced courses in Numerical Analysis, as typically taught in graduate courses in American and French universities. For example, it is the author’s experience that a one-semester course (on a three-hour per week basis) can be taught from Chapters 1, 2 and 3 (with the exception of Section 3.3), while another one-semester course can be taught from Chapters 4 and 6.

On the other hand, it is hoped that this book will prove to be useful for researchers interested in advanced aspects of the numerical analysis of the finite element method. In this respect, Section 3.3, Chapters 5, 7 and 8, and the sections on “Additional Bibliography and Comments” should provide many suggestions for conducting seminars.

Product Details

ISBN-13:	9780080875255
Publisher:	Elsevier Science
Publication date:	01/01/1978
Series:	ISSN , #4
Sold by:	Barnes & Noble
Format:	eBook
Pages:	529
File size:	13 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface to the Classics Edition	xv
	Preface	xix
	General plan and interdependence table	xxvi
1.	Elliptic boundary value problems	1
	Introduction	1
1.1.	Abstract problems	2
	The symmetric case. Variational inequalities	2
	The nonsymmetric case. The Lax-Milgram lemma	7
	Exercises	9
1.2.	Examples of elliptic boundary value problems	10
	The Sobolev spaces H[superscript m] ([Omega]). Green's formulas	10
	First examples of second-order boundary value problems	15
	The elasticity problem	23
	Examples of fourth-order problems: The biharmonic problem, the plate problem	28
	Exercises	32
	Bibliography and Comments	35
2.	Introduction to the finite element method	36
	Introduction	36
2.1.	Basic aspects of the finite element method	37
	The Galerkin and Ritz methods	37
	The three basic aspects of the finite element method. Conforming finite element methods	38
	Exercises	43
2.2.	Examples of finite elements and finite element spaces	43
	Requirements for finite element spaces	43
	First examples of finite elements for second order problems: n-Simplices of type (k), (3')	44
	Assembly in triangulations. The associated finite element spaces	51
	n-Rectangles of type (k). Rectangles of type (2'), (3'). Assembly in triangulations	55
	First examples of finite elements with derivatives as degrees of freedom: Hermite n-simplices of type (3), (3'). Assembly in triangulations	64
	First examples of finite elements for fourth-order problems: the Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations	69
	Exercises	77
2.3.	General properties of finite elements and finite element spaces	78
	Finite elements as triples (K, P, [Sigma]). Basic definitions. The P-interpolation operator	78
	Affine families of finite elements	82
	Construction of finite element spaces X[subscript h]. Basic definitions. The X[subscript h]-interpolation operator	88
	Finite elements of class l[superscript 0] and l[superscript 1]	95
	Taking into account boundary conditions. The spaces X[subscript 0h] and X[subscript 00h]	96
	Final comments	99
	Exercises	101
2.4.	General considerations on convergence	103
	Convergent family of discrete problems	103
	Cea's lemma. First consequences. Orders of convergence	104
	Bibliography and comments	106
3.	Conforming finite element methods for second order problems	110
	Introduction	110
3.1.	Interpolation theory in Sobolev spaces	112
	The Sobolev spaces W[superscript m,p]([Omega]). The quotient space W[superscript k+1,p]([Omega])/P[subscript k]([Omega])	112
	Error estimates for polynomial preserving operators	116
	Estimates of the interpolation errors |v - [Pi subscript K]v|[subscript m,q,K] for affine families of finite elements	122
	Exercises	126
3.2.	Application to second-order problems over polygonal domains	131
	Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]	131
	Sufficient conditions for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 1,[Omega] = 0	134
	Estimate of the error	136
	Concluding remarks. Inverse inequalities	139
	Exercises	143
3.3.	Uniform convergence	147
	A model problem. Weighted semi-norms |.|[subscript [phi],m,[Omega]	147
	Uniform boundedness of the mapping u [right arrow] u[subscript h] with respect to appropriate weighted norms	155
	Estimates of the errors	163
	Exercises	167
	Bibliography and comments	168
4.	Other finite element methods for second-order problems	174
	Introduction	174
4.1.	The effect of numerical integration	178
	Taking into account numerical integration. Description of the resulting discrete problem	178
	Abstract error estimate: The first Strang lemma	185
	Sufficient conditions for uniform V[subscript h]-ellipticity	187
	Consistency error estimates. The Bramble-Hilbert lemma	190
	Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]	199
	Exercises	201
4.2.	A nonconforming method	207
	Nonconforming methods for second-order problems. Description of the resulting discrete problem	207
	Abstract error estimate: The second Strang lemma	209
	An example of a nonconforming finite element: Wilson's brick	211
	Consistency error estimate. The bilinear lemma	217
	Estimate of the error ([Sigma subscript K[set membership]t subscript h]	220
	Exercises	223
4.3.	Isoparametric finite elements	224
	Isoparametric families of finite elements	224
	Examples of isoparametric finite elements	227
	Estimates of the interpolation errors |v - [Pi subscript K]v|[subscript m,q,K]	230
	Exercises	243
4.4.	Application to second order problems over curved domains	248
	Approximation of a curved boundary with isoparametric finite elements	248
	Taking into account isoparametric numerical integration. Description of the resulting discrete problem	252
	Abstract error estimate	255
	Sufficient conditions for uniform V[subscript h]-ellipticity	257
	Interpolation error and consistency error estimates	260
	Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]h]	266
	Exercises	270
	Bibliography and comments	272
	Additional bibliography and comments	276
	Problems on unbounded domains	276
	The Stokes problem	280
	Eigenvalue problems	283
5.	Application of the finite element method to some nonlinear problems	287
	Introduction	287
5.1.	The obstacle problem	289
	Variational formulation of the obstacle problem	289
	An abstract error estimate for variational inequalities	291
	Finite element approximation with triangles of type (1). Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]	294
	Exercises	297
5.2.	The minimal surface problem	301
	A formulation of the minimal surface problem	301
	Finite element approximation with triangles of type (1). Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]h]	302
	Exercises	310
5.3.	Nonlinear problems of monotone type	312
	A minimization problem over the space W[superscript 1,p subscript 0]([Omega]), 2 [less than or equal] p, and its finite element approximation with n-simplices of type (1)	312
	Sufficient condition for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 1,p,[Omega] = 0	317
	The equivalent problem Au = f. Two properties of the operator A	318
	Strongly monotone operators. Abstract error estimate	321
	Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,p,[Omega]	324
	Exercises	324
	Bibliography and comments	325
	Additional bibliography and comments	330
	Other nonlinear problems	330
	The Navier-Stokes problem	331
6.	Finite element methods for the plate problem	333
	Introduction	333
6.1.	Conforming methods	334
	Conforming methods for fourth-order problems	334
	Almost-affine families of finite elements	335
	A "polynomial" finite element of class l[superscript 1]: The Argyris triangle	336
	A composite finite element of class l[superscript 1]: The Hsieh-Clough-Tocher triangle	340
	A singular finite element of class l[superscript 1]: The singular Zienkiewicz triangle	347
	Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 2,[Omega]	352
	Sufficient conditions for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 2,[Omega] = 0	354
	Conclusions	354
	Exercises	356
6.2.	Nonconforming methods	362
	Nonconforming methods for the plate problem	362
	An example of a nonconforming finite element: Adini's rectangle	364
	Consistency error estimate. Estimate of the error ([Sigma subscript K[set membership]t subscript h]	367
	Further results	373
	Exercises	374
	Bibliography and comments	376
7.	A mixed finite element method	381
	Introduction	381
7.1.	A mixed finite element method for the biharmonic problem	383
	Another variational formuiation of the biharmonic problem	383
	The corresponding discrete problem. Abstract error estimate	386
	Estimate of the error (	390
	Concluding remarks	391
	Exercise	392
7.2.	Solution of the discrete problem by duality techniques	395
	Replacement of the constrained minimization problem by a saddlepoint problem	395
	Use of Uzawa's method. Reduction to a sequence of discrete Dirichlet problems for the operator - [Delta]	399
	Convergence of Uzawa's method	402
	Concluding remarks	403
	Exercises	404
	Bibliography and comments	406
	Additional bibliography and comments	407
	Primal, dual and primal-dual formulations	407
	Displacement and equilibrium methods	412
	Mixed methods	414
	Hybrid methods	417
	An attempt of general classification of finite element methods	421
8.	Finite element methods for shells	425
	Introduction	425
8.1.	The shell problem	426
	Geometrical preliminaries. Koiter's model	426
	Existence of a solution. Proof for the arch problem	431
	Exercises	437
8.2.	Conforming methods	439
	The discrete problem. Approximation of the geometry. Approximation of the displacement	439
	Finite element methods conforming for the displacements	440
	Consistency error estimates	443
	Abstract error estimate	447
	Estimate of the error ([Sigma superscript 2 subscript [alpha] = 1] [double vertical line]u[subscript [alpha] - u[subscript [alpha]h double vertical line superscript 2 subscript 1,[Omega] + [double vertical line]u[subscript 3] - u[subscript 3h double vertical line superscript 2 subscript 2,[Omega])[superscript 1/2]	448
	Finite element methods conforming for the geometry	450
	Conforming finite element methods for shells	450
8.3.	A nonconforming method for the arch problem	451
	The circular arch problem	451
	A natural finite element approximation	452
	Finite element methods conforming for the geometry	453
	A finite element method which is not conforming for the geometry. Definition of the discrete problem	453
	Consistency error estimates	461
	Estimate of the error (	465
	Exercise	466
	Bibliography and comments	466
	Epilogue: Some "real-life" finite element model examples	469
	Bibliography	481
	Glossary of symbols	512
	Index	521

Editorial Reviews

A textbook for a two-semester graduate course in numerical analysis. Students are assumed to have a good knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Ciarlet (City U. of Hong Kong, Kowloon) adds a new preface to the 1978 edition, published by North-Holland, which is cited in Books for College Libraries, 3rd ed. Annotation c. Book News, Inc., Portland, OR