The Finite Element Method for Elliptic Problems,9780898715149

The Finite Element Method for Elliptic Problems

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ISBN13: 9780898715149
ISBN10: 0898715148
Format: Paperback
Pub. Date: 2002-04-01
Publisher(s): Society for Industrial & Applied
List Price: $93.58

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Summary

The Finite Element Method for Elliptic Problems is the only book available that fully analyzes 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, and also a working textbook for graduate courses in numerical analysis. It includes many useful figures, as well as 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. As such the author has provided a bibliography of recent texts that complement the classic material in these chapters.

Author Biography

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

Table of Contents

Preface to the Classics Edition xv
Preface xix
General Plan and Interdependence Table xxvi
Elliptic Boundary Value Problems
1(35)
Introduction
1(1)
Abstract problems
2(8)
The symmetric case. Variational inequalities
2(5)
The nonsymmetric case. The Lax-Milgram lemma
7(2)
Exercises
9(1)
Examples of elliptic boundary value problems
10(26)
The Sobolev spaces Hm(Ω). Green's formulas
10(5)
First examples of second-order boundary value problems
15(8)
The elasticity problem
23(5)
Examples of fourth-order problems: The biharmonic problem, the plate problem
28(4)
Exercises
32(3)
Bibliography and Comments
35(1)
Introduction to the Finite Element Method
36(74)
Introduction
36(1)
Basic aspects of the finite element method
37(6)
The Galerkin and Ritz methods
37(1)
The three basic aspects of the finite element method. Conforming finite element methods
38(5)
Exercises
43(1)
Examples of finite elements and finite element spaces
43(35)
Requirements for finite element spaces
43(1)
First examples of finite elements for second order problems: n-Simplices of type (k), (3')
44(7)
Assembly in triangulations. The associated finite element spaces
51(4)
n-Rectangles of type (k). Rectangles of type (2'), (3'). Assembly in triangulations
55(9)
First examples of finite elements with derivatives as degrees of freedom: Hermite n-simplices of type (3), (3'). Assembly in triangulations
64(5)
First examples of finite elements for fourth-order problems: the Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations
69(8)
Exercises
77(1)
General properties of finite elements and finite element spaces
78(25)
Finite elements as triples (K, P, Σ). Basic definitions. The P-inter-polation operator
78(4)
Affine families of finite elements
82(6)
Construction of finite element spaces Xh. Basic definitions. The Xh - interpolation operator
88(7)
Finite elements of class l0 and l1
95(1)
Taking into account boundary conditions. The spaces X0h and X00h
96(3)
Final comments
99(2)
Exercises
101(2)
General considerations on convergence
103(7)
Convergent family of discrete problems
103(1)
Cea's lemma. First consequences. Orders of convergence
104(2)
Bibliography and comments
106(4)
Conforming Finite Element Methods for Second Order Problems
110(64)
Introduction
110(2)
Interpolation theory in Sobolev spaces
112(19)
The Sobolev spaces Wm,p(Ω). The quotient space Wk+1,p(Ω)/Pk(Ω)
112(4)
Error estimates for polynomial preserving operators
116(6)
Estimates of the interpolation errors |υ - Πk&υ|m,q,k for affine families of finite elements
122(4)
Exercises
126(5)
Application to second-order problems over polygonal domains
131(16)
Estimate of the error ||u - uh||l,Ω
131(3)
Sufficient conditions for limh→0||u - uh||i Ω = 0
134(2)
Estimate of the error |u - uh|0Ω. The Aubin-Nitsche lemma
136(3)
Concluding remarks. Inverse inequalities
139(4)
Exercises
143(4)
Uniform convergence
147(27)
A model problem. Weighted semi-norms |·|φ,m,Ω
147(8)
Uniform boundedness of the mapping u → uh with respect to appropriate weighted norms
155(8)
Estimates of the errors |u - uh|o,∞,Ω and |u - uh|l,∞Ω. Nitsche's method of weighted norms
163(4)
Exercises
167(1)
Bibliography and comments
168(6)
Other Finite Element Methods for Second-Order Problems
174(113)
Introduction
174(4)
The effect of numerical integration
178(29)
Taking into account numerical integration. Description of the resulting discrete problem
178(7)
Abstract error estimate: The first Strang lemma
185(2)
Sufficient conditions for uniform Vh-ellipticity
187(3)
Consistency error estimates. The Bramble-Hilbert lemma
190(9)
Estimate of the error ||u - uh||l,Ω
199(2)
Exercises
201(6)
A nonconforming method
207(17)
Nonconforming methods for second-order problems. Description of the resulting discrete problem
207(2)
Abstract error estimate: The second Strang lemma
209(2)
An example of a nonconforming finite element: Wilson's brick
211(6)
Consistency error estimate. The bilinear lemma
217(3)
Estimate of the error (Sigma;kεjh|u - uh|21,k)1/2
220(3)
Exercises
223(1)
Isoparametric finite elements
224(24)
Isoparametric families of finite elements
224(3)
Examples of isoparametric finite elements
227(3)
Estimates of the interpolation errors |υ - uh||1.Ωh
230(13)
Exercises
243(5)
Application to second order problems over curved domains
248(39)
Approximation of a curved boundary with isoparametric finite elements
248(4)
Taking into account isoparametric numerical integration. Description of the resulting discrete problem
252(3)
Abstract error estimate
255(2)
Sufficient conditions for uniform Vh-ellipticity
257(3)
Interpolation error and consistency error estimates
260(6)
Estimate of the error ||u - uh||lΩh
266(4)
Exercises
270(2)
Bibliography and comments
272(4)
Additional bibliography and comments
276(1)
Problems on unbounded domains
276(4)
The Stokes problem
280(3)
Eigenvalue problems
283(4)
Application of the Finite Element Method to Some Nonlinear Problems
287(46)
Introduction
287(2)
The obstacle problem
289(12)
Variational formulation of the obstacle problem
289(2)
An abstract error estimate for variational inequalities
291(3)
Finite element approximation with triangles of type (1). Estimate of the error ||u - uh||lΩ
294(3)
Exercises
297(4)
The minimal surface problem
301(11)
A formulation of the minimal surface problem
301(1)
Finite element approximation with triangles of type (1). Estimate of the error ||u - uj||lΩh
302(8)
Exercises
310(2)
Nonlinear problems of monotone type
312(21)
A minimization problem over the space W01,p (Omega;), 2 ≤ p, and its finite element approximation with n-simplices of type (1)
312(5)
Sufficient condition for lim h→o||u - uh||l,p,Ω=0
317(1)
The equivalent problem Au = f. Two properties of the operator A
318(3)
Strongly monotone operators. Abstract error estimate
321(3)
Estimate of the error ||u - uh||l,p,Ω
324(1)
Exercises
324(1)
Bibliography and comments
325(5)
Additional bibliography and comments
330(1)
Other nonlinear problems
330(1)
The Navier-Stokes problem
331(2)
Finite Element Methods for the Plate Problem
333(48)
Introduction
333(1)
Conforming methods
334(28)
Conforming methods for fourth-order problems
334(1)
Almost-affine families of finite elements
335(1)
A ``polynomial'' finite element of class l1: The Argyris triangle
336(4)
A composite finite element of class l1: The Hsieh-Clough-Tocher triangle
340(7)
A singular finite element of class l1: The singular Zienkiewicz triangle
347(5)
Estimate of the error ||u - uh||2,Ω
352(2)
Sufficient conditions for lim h→o||u - uh||2,Ω = 0
354(1)
Conclusions
354(2)
Exercises
356(6)
Nonconforming methods
362(19)
Nonconforming methods for the plate problem
362(2)
An example of a nonconforming finite element: Adini's rectangle
364(3)
Consistency error estimate. Estimate of the error (Sigma;kεjh|uh|22.k)1/2
367(6)
Further results
373(1)
Exercises
374(2)
Bibliography and comments
376(5)
A Mixed Finite Element Method
381(44)
Introduction
381(2)
A mixed finite element method for the biharmonic problem
383(12)
Another variational formulation of the biharmonic problem
383(3)
The corresponding discrete problem. Abstract error estimate
386(4)
Estimate of the error (|u - uh|l,Ω + |Δu + φh|o,Ω
390(1)
Concluding remarks
391(1)
Exercise
392(3)
Solution of the discrete problem by duality techniques
395(30)
Replacement of the constrained minimization problem by a saddle-point problem
395(4)
Use of Uzawa's method. Reduction to a sequence of discrete Dirichlet problems for the operator - Δ
399(3)
Convergence of Uzawa's method
402(1)
Concluding remarks
403(1)
Exercises
404(2)
Bibliography and comments
406(1)
Additional bibliography and comments
407(1)
Primal, dual and primal-dual formulations
407(5)
Displacement and equilibrium methods
412(2)
Mixed methods
414(3)
Hybrid methods
417(4)
An attempt of general classification of finite element methods
421(4)
Finite Element Methods for Shells
425(44)
Introduction
425(1)
The shell problem
426(13)
Geometrical preliminaries. Koiter's model
426(5)
Existence of a solution. Proof for the arch problem
431(6)
Exercises
437(2)
Conforming methods
439(12)
The discrete problem. Approximation of the geometry. Approximation of the displacement
439(1)
Finite element methods conforming for the displacements
440(3)
Consistency error estimates
443(4)
Abstract error estimate
447(1)
Estimate of the error (Σ 2 α = 1 ||uα - uα||21Ω+||u3 - u3h||22Ω)1/2
448(2)
Finite element methods conforming for the geometry
450(1)
Conforming finite element methods for shells
450(1)
A nonconforming method for the arch problem
451(18)
The circular arch problem
451(1)
A natural finite element approximation
452(1)
Finite element methods conforming for the geometry
453(1)
A finite element method which is not conforming for the geometry. Definition of the discrete problem
453(8)
Consistency error estimates
461(4)
Estimate of the error (|u1 - u1h|21.l+|u2 - h|22.l)1/2
465(1)
Exercise
466(1)
Bibliography and comments
466(3)
Epilogue: Some ``real-life'' finite element model examples 469(12)
Bibliography 481(31)
Glossary of Symbols 512(9)
Index 521

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