The finite element method for elliptic problems

This is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty.

Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters.

Author Notes

Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences

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

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