Mixed finite element methods and applications
 Responsibility
 Daniele Boffi, Franco Brezzi, Michel Fortin.
 Language
 English.
 Imprint
 Berlin ; Heidelberg : Springer, c2013.
 Physical description
 xiv, 685 p. : ill. ; 25 cm.
 Series
 Springer series in computational mathematics ; 44.
Access
Creators/Contributors
 Author/Creator
 Boffi, Daniele.
 Contributor
 Brezzi, F. (Franco), 1945
 Fortin, Michel, 1945
Contents/Summary
 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.
 Publisher's Summary
 Nonstandard finite element methods, in particular mixed methods, are central to many applications. In this text the authors, Boffi, Brezzi and Fortin present a general framework, starting with a finite dimensional presentation, then moving on to formulation in Hilbert spaces and finally considering approximations, including stabilized methods and eigenvalue problems. This book also provides an introduction to standard finite element approximations, followed by the construction of elements for the approximation of mixed formulations in H(div) and H(curl). The general theory is applied to some classical examples: Dirichlet's problem, Stokes' problem, plate problems, elasticity and electromagnetism.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Series
 Springer series in computational mathematics, 01793632 ; 44
 Available in another form
 9783642365195 (online)
 (GyWOH)har135015712
 ISBN
 9783642365188 (hd.bd.)
 9783642365195 (online)
 3642365183