| Preface to the Classics Edition |
| Preface |
| General plan and interdependence table |
| Elliptic boundary value problems / 1.： |
| Introduction |
| Abstract problems / 1.1.： |
| The symmetric case. Variational inequalities |
| The nonsymmetric case. The Lax-Milgram lemma |
| Exercises |
| Examples of elliptic boundary value problems / 1.2.： |
| The Sobolev spaces H[superscript m] ([Omega]). Green's formulas |
| First examples of second-order boundary value problems |
| The elasticity problem |
| Examples of fourth-order problems: The biharmonic problem, the plate problem |
| Bibliography and Comments |
| Introduction to the finite element method / 2.： |
| Basic aspects of the finite element method / 2.1.： |
| The Galerkin and Ritz methods |
| The three basic aspects of the finite element method. Conforming finite element methods |
| Examples of finite elements and finite element spaces / 2.2.： |
| Requirements for finite element spaces |
| First examples of finite elements for second order problems: n-Simplices of type (k), (3') |
| Assembly in triangulations. The associated finite element spaces |
| n-Rectangles of type (k). Rectangles of type (2'), (3'). Assembly in triangulations |
| First examples of finite elements with derivatives as degrees of freedom: Hermite n-simplices of type (3), (3'). Assembly in triangulations |
| First examples of finite elements for fourth-order problems: the Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations |
| General properties of finite elements and finite element spaces / 2.3.： |
| Finite elements as triples (K, P, [Sigma]). Basic definitions. The P-interpolation operator |
| Affine families of finite elements |
| Construction of finite element spaces X[subscript h]. Basic definitions. The X[subscript h]-interpolation operator |
| Finite elements of class l[superscript 0] and l[superscript 1] |
| Taking into account boundary conditions. The spaces X[subscript 0h] and X[subscript 00h] |
| Final comments |
| General considerations on convergence / 2.4.： |
| Convergent family of discrete problems |
| Cea's lemma. First consequences. Orders of convergence |
| Bibliography and comments |
| Conforming finite element methods for second order problems / 3.： |
| Interpolation theory in Sobolev spaces / 3.1.： |
| The Sobolev spaces W[superscript m,p]([Omega]). The quotient space W[superscript k+1,p]([Omega])/P[subscript k]([Omega]) |
| Error estimates for polynomial preserving operators |
| Estimates of the interpolation errors |v - [Pi subscript K]v|[subscript m,q,K] for affine families of finite elements |
| Application to second-order problems over polygonal domains / 3.2.： |
| Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega] |
| Sufficient conditions for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 1,[Omega] = 0 |
| Estimate of the error |
| Concluding remarks. Inverse inequalities |
| Uniform convergence / 3.3.： |
| A model problem. Weighted semi-norms |.|[subscript [phi],m,[Omega] |
| Uniform boundedness of the mapping u [right arrow] u[subscript h] with respect to appropriate weighted norms |
| Estimates of the errors |
| Other finite element methods for second-order problems / 4.： |
| The effect of numerical integration / 4.1.： |
| Taking into account numerical integration. Description of the resulting discrete problem |
| Abstract error estimate: The first Strang lemma |
| Sufficient conditions for uniform V[subscript h]-ellipticity |
| Consistency error estimates. The Bramble-Hilbert lemma |
| A nonconforming method / 4.2.： |
| Nonconforming methods for second-order problems. Description of the resulting discrete problem |
| Abstract error estimate: The second Strang lemma |
| An example of a nonconforming finite element: Wilson's brick |
| Consistency error estimate. The bilinear lemma |
| Estimate of the error ([Sigma subscript K[set membership]t subscript h] |
| Isoparametric finite elements / 4.3.： |
| Isoparametric families of finite elements |
| Examples of isoparametric finite elements |
| Estimates of the interpolation errors |v - [Pi subscript K]v|[subscript m,q,K] |
| Application to second order problems over curved domains / 4.4.： |
| Approximation of a curved boundary with isoparametric finite elements |
| Taking into account isoparametric numerical integration. Description of the resulting discrete problem |
| Abstract error estimate |
| Interpolation error and consistency error estimates |
| Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,[Omega]h] |
| Additional bibliography and comments |
| Problems on unbounded domains |
| The Stokes problem |
| Eigenvalue problems |
| Application of the finite element method to some nonlinear problems / 5.： |
| The obstacle problem / 5.1.： |
| Variational formulation of the obstacle problem |
| An abstract error estimate for variational inequalities |
| 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] |
| The minimal surface problem / 5.2.： |
| A formulation of the minimal surface problem |
| 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] |
| Nonlinear problems of monotone type / 5.3.： |
| 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) |
| Sufficient condition for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 1,p,[Omega] = 0 |
| The equivalent problem Au = f. Two properties of the operator A |
| Strongly monotone operators. Abstract error estimate |
| Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 1,p,[Omega] |
| Other nonlinear problems |
| The Navier-Stokes problem |
| Finite element methods for the plate problem / 6.： |
| Conforming methods / 6.1.： |
| Conforming methods for fourth-order problems |
| Almost-affine families of finite elements |
| A "polynomial" finite element of class l[superscript 1]: The Argyris triangle |
| A composite finite element of class l[superscript 1]: The Hsieh-Clough-Tocher triangle |
| A singular finite element of class l[superscript 1]: The singular Zienkiewicz triangle |
| Estimate of the error [double vertical line]u - u[subscript h double vertical line subscript 2,[Omega] |
| Sufficient conditions for lim[subscript h[right arrow]0 double vertical line]u - u[subscript h double vertical line subscript 2,[Omega] = 0 |
| Conclusions |
| Nonconforming methods / 6.2.： |
| Nonconforming methods for the plate problem |
| An example of a nonconforming finite element: Adini's rectangle |
| Consistency error estimate. Estimate of the error ([Sigma subscript K[set membership]t subscript h] |
| Further results |
| A mixed finite element method / 7.： |
| A mixed finite element method for the biharmonic problem / 7.1.： |
| Another variational formuiation of the biharmonic problem |
| The corresponding discrete problem. Abstract error estimate |
| Estimate of the error ( |
| Concluding remarks |
| Exercise |
| Solution of the discrete problem by duality techniques / 7.2.： |
| Replacement of the constrained minimization problem by a saddlepoint problem |
| Use of Uzawa's method. Reduction to a sequence of discrete Dirichlet problems for the operator - [Delta] |
| Convergence of Uzawa's method |
| Primal, dual and primal-dual formulations |
| Displacement and equilibrium methods |
| Mixed methods |
| Hybrid methods |
| An attempt of general classification of finite element methods |
| Finite element methods for shells / 8.： |
| The shell problem / 8.1.： |
| Geometrical preliminaries. Koiter's model |
| Existence of a solution. Proof for the arch problem |
| The discrete problem. Approximation of the geometry. Approximation of the displacement / 8.2.： |
| Finite element methods conforming for the displacements |
| Consistency error estimates |
| 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] |
| Finite element methods conforming for the geometry |
| Conforming finite element methods for shells |
| A nonconforming method for the arch problem / 8.3.： |
| The circular arch problem |
| A natural finite element approximation |
| A finite element method which is not conforming for the geometry. Definition of the discrete problem |
| Epilogue: Some "real-life" finite element model examples |
| Bibliography |
| Glossary of symbols |
| Index |
| Preface to the Classics Edition |
| Preface |
| General plan and interdependence table |
図書
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