The Finite Element Method for Elliptic Problems

by Ciarlet, Philippe G.

• ISBN13: 9780898715149
• ISBN10: 0898715148
• Format: Paperback
• Copyright: 2002-04-01
• Publisher: Society for Industrial & Applied

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