jump to search box

Mixed finite element methods and applications / Daniele Boffi, Franco Brezzi, Michel Fortin.

Availability

Online

At the Library

Other libraries

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
  • Raviart-Thomas 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 vector-valued 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 two-dimensional case
  • Basis functions for H(div : K) : the three-dimensional case
  • Basis functions for H(curl : K) : the two-dimensional case
  • Basis functions for H(curl : K) : the three-dimensional 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 inf-sup condition for the matrix b : an elementary discussion
  • The inf-sup 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 inf-sup 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 inf-sup 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 inf-sup condition : criteria
  • Some linguistic considerations
  • General considerations
  • The inf-sup condition and the B-compatible 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 inf-sup 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 "element-wise 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
  • Non-standard 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 macro-hybrid methods and domain decompositions
  • Dual hybrid methods
  • Numerical solutions
  • Preliminaries
  • Inter-element 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 two-dimensional cases
  • The two-dimensional case
  • The three-dimensional 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 P1-P0 Approximation
  • Building a B-compatible operator : the simplest stable elements
  • Building a B-compatible operator
  • A stable case : the mini element
  • Another stable approximation : the bi-dimensional P2-P0 element
  • The nonconforming P1-P0 approximation
  • Other techniques for checking the inf-sup condition
  • Projection onto constants
  • Verfürth's trick
  • Space and domain decomposition techniques
  • Macro-element technique
  • Making use of the internal degrees of freedom
  • Two-dimensional stable elements
  • Continuous pressure elements
  • Discontinuous pressure elements
  • Quadrilateral elements, Qk-Pk-1 elements
  • Three-dimensional stable elements
  • Continuous pressure 3-d elements
  • Discontinuous pressure 3-d elements
  • Pk-Pk-1 schemes and generalised Hood-Taylor elements
  • Discontinuous pressure Pk-Pk-1 elements
  • Generalised Hood-Taylor elements
  • Other developments for divergence-free stokes approximation and mass conservation
  • Exactly divergence-free stokes elements, discontinuous Galerkin methods
  • Stokes elements allowing for element-wise mass conservation
  • Spurious pressure modes
  • Living with spurious pressure modes : partial convergence
  • The bilinear velocity-constant pressure Q1-P0 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 Sh-1
  • 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 relaxed-symmetry 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 fourth-order 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 Mindlin-Reissner 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 two-dimensional case
  • Discrete compactness property
  • Nodal finite elements
  • Edge finite elements
  • Enforcing the divergence-free 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, 0179-3632 ; 44
Springer series in computational mathematics ; 44.
Subjects:
ISBN:
9783642365188
3642365183

powered by Blacklight
© Stanford University. Stanford, California 94305. (650) 725-1064. Terms of Use | Copyright Complaints | Opt Out of Analytics
jump to top