| Description |
1 online resource. |
|
text txt rdacontent |
|
computer c rdamedia |
|
online resource cr rdacarrier |
| Series |
Springer series in computational mathematics ; 44.
|
| Note |
Description based on print version record. |
| 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. |
| Reproduction |
Electronic reproduction. Ann Arbor, MI Available via World Wide Web. |
| Note |
Description based on print version record. |
| Subject |
Finite element method.
|
|
Finite element method -- Data processing.
|
| Added Author |
Fortin, Michel, 1945- author.
|
|
Brezzi, F. (Franco), 1945- author.
|
|
ProQuest (Firm)
|
| Related To |
Printed edition: 9783642365188 |
|
Original 3642365183 9783642365188 |
| ISBN |
9783642365195 (electronic bk.) |
|
3642365191 (electronic bk.) |
|
3642365183 |
|
9783642365188 |
| UPC # |
10.1007/978-3-642-36519-5 doi |
| OCLC # |
EBC1317730 |
|
