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TA347 .F5 B64 2013
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Mixed finite element methods and applications / Daniele Boffi, Franco Brezzi, Michel Fortin.
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TA347 .F5 B64 2013
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TA347 .F5 B64 2013
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Author/Creator:
Boffi, Daniele.
Language:
English.
Publication date:
2013
Imprint:
Berlin ; Heidelberg : Springer, c2013.
Format:
Book
xiv, 685 p. : ill. ; 25 cm.
Bibliography:
Includes bibliographical references and index.
Contents:
Variational formulations and finite element methods
Classical methods
Model problems and elementary properties of some functional spaces
Eigenvalue problems
Duality methods
Generalities
Examples for symmetric problems
Duality methods for non symmetric bilinear forms
Mixed eigenvalue problems
Domain decomposition methods, hybrid methods
Modified variational formulations
Augmented formulations
Perturbed formulations
Bibliographical remarks
Function spaces and finite element approximations
Properties of the spaces Hm(...), H(div ; ...), and H(curl : ...)
Basic properties
Properties relative to a partition of ...
Properties relative to a change of variables
De Rham diagram
Finite element approximations of H1(...) and H2(...)
Conforming methods
Explicit basis functions on triangles and tetrahedra
Nonconforming methods
Quadrilateral finite elements on non affine meshes
Quadrilateral approximation of scalar functions
Non polynomial approximations
Scaling arguments
Simplicial approximations of H(div : ...) and H(curl : ...)
Simplicial approximations of H(div : ...)
Simplicial approximation of H(curl : ...)
Approximations of H(div : K) on rectangles and cubes
RaviartThomas elements on rectangles and cubes
Other approximations of H(div : K) on rectangles
Other approximations of H(div : K) on cubes
Approximations of H(curl : K) on cubes
Interpolation operator and error estimates
Approximations of H(div : K)
Approximation spaces for H(div : ...)
Approximations of H(curl : ...)
Approximation spaces for H(curl : ...)
Quadrilateral and hexahedral approximation of vectorvalued functions in H(div : ...) and H(curl : ...)
Discrete exact sequences
Explicit basis functions for H(div : K) and H(curl : K) on triangles and tetrahedra
Basis functions for H(div : K) : the twodimensional case
Basis functions for H(div : K) : the threedimensional case
Basis functions for H(curl : K) : the twodimensional case
Basis functions for H(curl : K) : the threedimensional case
Concluding remarks
Algebraic aspects of saddle point problems
Notation, and basic results in linear algebra
Basic definitions
Subspaces
Orthogonal subspaces
Orthogonal projections
Basic results
Restrictions of operators
Existence and uniqueness of solutions : the solvability problem
A preliminary discussion
The necessary and sufficient condition
Sufficient conditions
Examples
Composite matrices
The solvability problem for perturbed matrices
Preliminary results
Main results
Examples
Stability
Assumptions on the norms
The infsup condition for the matrix b : an elementary discussion
The infsup condition and the singular values
The case of A elliptic on the whole space
The case of A elliptic on the kernel of B
The case of A satisfying an infsup on the kernel of B
Additional results
Some necessary conditions
The case of B not surjective : modifikation of the problem
Some special cases
Composite matrices
Stability of perturbed matrices
The basic estimate
The symmetric case for perturbed matrices
Saddle point problems in hilbert spaces
Reminders on hilbert spaces
Scalar products, norms, completeness
Closed subspaces and dense subspaces
Orthogonality
Continuous linear operators, dual spaces, polar spaces
Bilinear forms and associated operators : transposed operators
Dual spaces of linear subspaces
Identification of a space with its dual space
Restrictions of operators to closed subspaces
Quotient spaces
Existence and uniqueness of solutions
Mixed formulations in Hilbert spaces
Stability constants and infsup conditions
The main result
The case of lmB ... Q'
Examples
Existence and uniqueness for perturbed problems
Regular perturbations
Singular perturbations
Approximation of saddle point problems
Basic results
The basic assumptions
The discrete operators
Error estimates for finite dimensional approximations
Discrete stability and error estimates
Additional error estimates for the basic problem
Variants of error estimates
A simple example
An important example : the pressure in the homogeneous stokes problem
The case of Ker Bth ... (0)
The case of Ker Bth ... Ker Bt
The case of Ker Bth ... Ker Bt
The case of ... going to zero
The infsup condition : criteria
Some linguistic considerations
General considerations
The infsup condition and the Bcompatible interpolation operator ...
Construction of ...
An alternative strategy : switching norms
Extensions of error estimates
Perturbed problems
Penalty methods
Singular perturbations
Nonconforming methods
Dual error estimates
Numerical properties of the discrete problem
The matrix form of the discrete problem
And if the infsup condition does not hold?
Solution methods
Concluding remarks
Complements : stabilisation methods, eigenvalue problems
Augmented formulations
An abstract framework for stabiiised methods
Stabilising terms
Stability conditions for augmented formulations
Discretisations of augmented formulations
Stabilising with the "elementwise equations"
Other stabilisations
General stability conditions
Stability of discretised formulations
Minimal stabilisations
Another form of minimal stabilisation
Enhanced strain methods
Eigenvalue problems
Some classical results
Eigenvalue problems in mixed form
Special results for problems of Type (f, 0) and (0, g)
Eigenvalue problems of the Type (o, g)
Eigenvalue problems of the Form (0, g)
Mixed methods for elliptic problems
Nonstandard methods for Dirichlet's problem
Description of the problem
Mixed finite element methods for Dirichlet's problem
Eigenvalue problem for the mixed formulation
Primal hybrid methods
Primal macrohybrid methods and domain decompositions
Dual hybrid methods
Numerical solutions
Preliminaries
Interelement multipliers
A brief analysis of the computational effort
Error analysis for the multiplier
Error estimates in other norms
Application to an equation arising from semiconductor theory
Using anisotropie meshes
Relations with finite volume methods
The one and twodimensional cases
The twodimensional case
The threedimensional case
Nonconforming methods : a trap to avoid
Augmented formulations (Galerkin least squares methods)
A posteriori error estimates
Incompressible materials and flow problems
Introduction
The stokes problem as a mixed problem
Mixed formulation
Some examples of failure and empirical cures
Continuous pressure : the ... P1 P1 Element
Discontinuous pressure : the P1P0 Approximation
Building a Bcompatible operator : the simplest stable elements
Building a Bcompatible operator
A stable case : the mini element
Another stable approximation : the bidimensional P2P0 element
The nonconforming P1P0 approximation
Other techniques for checking the infsup condition
Projection onto constants
Verfürth's trick
Space and domain decomposition techniques
Macroelement technique
Making use of the internal degrees of freedom
Twodimensional stable elements
Continuous pressure elements
Discontinuous pressure elements
Quadrilateral elements, QkPk1 elements
Threedimensional stable elements
Continuous pressure 3d elements
Discontinuous pressure 3d elements
PkPk1 schemes and generalised HoodTaylor elements
Discontinuous pressure PkPk1 elements
Generalised HoodTaylor elements
Other developments for divergencefree stokes approximation and mass conservation
Exactly divergencefree stokes elements, discontinuous Galerkin methods
Stokes elements allowing for elementwise mass conservation
Spurious pressure modes
Living with spurious pressure modes : partial convergence
The bilinear velocityconstant pressure Q1P0 element
Eigenvalue problems
Nearly incompressible elasticity, reduced integration methods and relation with penalty methods
Variational formulations and admissible discretisations
Reduced integration methods
Effects of inexact integration
Other stabilisation procedures
Augmented method for the stokes problem
Defining an approximate inverse Sh1
Minimal stabilisations for stokes
Concluding remarks : choice of elements
Choice of elements
Complements on elasticity problems
Introduction
Continuous formulation of Stress methods
Numerical approximations of Stress formulations
Relaxed symmetry
Tensors, tensorial notation and results on symmetry
Continuous formulation of the relaxed symmetry approach
Numerical approximation of relaxedsymmetry formulations
Some families of methods with reduced symmetry
Methods based on stokes elements
Stabilisation by H(curl) bubbles
Two examples
Methods based on the properties of ...
Loosing the inclusion of kernel : stabiiised methods
Concluding remarks
Complements on plate problems
A mixed fourthorder problem
The ... biharmonic problem
Eigenvalues of the biharmonic problem
Dual hybrid methods for plate bending problems
Mixed methods for linear thin plates
Moderately thick plates
Generalities
The mathematical formulation
Mixed formulation of the MindlinReissner model
A decomposition principle and the stokes
Connection
Discretisation of the problem
Continuous pressure approximations
Discontinuous pressure elements
Mixed finite elements for electromagnetic problems
Useful results about the space H(curl : ...), its boundary traces, and the de Rham complex
The de Rham complex and the Helmholtz decomposition when ... is simply connected
The Friedrichs inequality
Extension to more general topologies
H(curl : ...) In two space dimensions
The time harmonic Maxwell system
Maxwell's eigenvalue problem
Analysis of the time harmonic Maxwell system
Approximation of the time harmonic Maxwell equations
Approximation of the Maxwell eigenvalue problem
Analysis of the twodimensional case
Discrete compactness property
Nodal finite elements
Edge finite elements
Enforcing the divergencefree condition by a penalty method
Some remarks on exterior calculus
Concluding remarks
References
Index.
Contributor:
Brezzi, F. (Franco), 1945
Fortin, Michel, 1945
Available in another form:
9783642365195 (online)
(GyWOH)har135015712
Series:
Springer series in computational mathematics, 01793632 ; 44
Springer series in computational mathematics ;
44.
Subjects:
Finite element method.
Finite element method
>
Data processing.
ISBN:
9783642365188
3642365183
Catkey: 10212832
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