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 |
Elliptic boundary value problems / 1.: |
Introduction |
Abstract problems / 1.1.: |