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1.

図書

図書
edited by Marian Neamtu, Larry Schumaker
出版情報: New York ; London : Springer, c2012  xvii, 415 p. ; 24 cm
シリーズ名: Springer proceedings in mathematics ; 13
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2.

図書

図書
edited by G.G. Lorentz, C.K. Chui, L.L. Schumaker
出版情報: New York : Academic Press, 1976  xi, 588 p. ; 24 cm
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3.

図書

図書
Dietrich Braess ; translated by Larry L. Schumaker
出版情報: Cambridge : Cambridge University Press, 1997  xvi, 323 p. ; 24 cm
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目次情報: 続きを見る
Preface to the Third English Edition page
Preface to the First English Edition
Preface to the German Edition
Notation
Introduction / Chapter I:
Examples and Classification of PDE's / 1:
Examples
Classification of PDE's
Well-posed Problems
Problems
The Maximum Principle / 2:
Corollaries
Problem
Finite Difference Methods / 3:
Discretization
Discrete maximum principle
A Convergence Theory for Difference Methods / 4:
Consistency
Local and global error
Limits of the convergence theory
Conforming Finite Elements / Chapter II:
Sobolev Spaces
Introduction to Sobolev spaces
Friedrichs' inequality
Possible singularities of H1 functions
Compact imbeddings
Variational Formulation of Elliptic Boundary-Value Problems of Second Order
Variational formulation
Reduction to homogeneous boundary conditions
Existence of solutions
Inhomogeneous boundary conditions
The Neumann Boundary-Value Problem. A Trace Theorem
Ellipticity in H1
Boundary-value problems with natural boundary conditions
Neumann boundary conditions
Mixed boundary conditions
Proof of the trace theorem
Practical consequences of the trace theorem
The Ritz-Galerkin Method and Some Finite Elements
Model Problem
Some Standard Finite Elements / 5:
Requirements on the meshes
Significance of the differentiability properties
Triangular elements with complete polynomials
Remarks on C1 elements
Bilinear elements
Quadratic rectangular elements
Affine families
Choice of an element
Approximation Properties / 6:
The Bramble-Hilbert lemma
Bilinear quadrilateral elements
Inverse estimates
Clément's interpolation
Appendix: On the optimality of the estimates
Error Bounds for Elliptic Problems of Second Order / 7:
Remarks on regularity
Error bounds in the energy norm
L2 estimates
A simple L? estimate
The L2-projector
Computational Considerations / 8:
Assembling the stiffness matrix
Static condensation
Complexity of setting up the matrix
Effect on the choice of a grid
Local mesh refinement
Implementation of the Neumann boundary-value Problem
Nonconforming and Other Methods / Chapter III:
Abstract Lemmas and a Simple Boundary Approximation
Generalizations of Céa's lemma
Duality methods
The Crouzeix-Raviart element
A Simple approximation to curved boundaries
Modifications of the duality argument
Isoparametric Elements
Isoparametric triangular elements
Isoparametric quadrilateral elements
Further Tools from Functional Analysis
Negative norms
Adjoint operators
An abstract existence theorem
An abstract convergence theorem
Proof of Theorem 3.4
Saddle Point Problems
Saddle points and minima
The inf-sup condition
Mixed finite element methods
Fortin interpolation
Saddle point problems with penalty term
Typical applications
Mixed Methods for the Poisson Equation
The Poisson equation as a mixed problem
The Raviart - Thomas element
Interpolation by Raviart-Thomas elements
Implementation and postprocessing
Mesh-dependent norms for the Raviart-Thomas element
The Softening behaviour of mixed methods
The Stokes Equation
Nearly incompressible flows
Finite Elements for the Stokes Problems
An instable element
The Taylor-Hood element
The MINI element
The divergence-free nonconforming P1 element
A Posteriori Error Estimates
Residual estimators
Lower estimates
Remark on other estimators
Local mesh refinement and convergence
A Posteriori Error Estimates via the Hypercircle Method / 9:
The Conjugate Gradient Method / Chapter IV:
Classical Iterative Methods for Solving Linear Systems
Stationary linear processes
The Jacobi and Gauss-Seidel methods
The model problem
Overrelaxation
Gradient Methods
The general gradient method
Gradient methods and quadratic functions
Convergence behavior in the case of large condition numbers
Conjugate Gradient and the Minimal Residual Method
The CG algorithm
Analysis of the CG method as an optimal method
The minimal residual method
Indefinite and unsymmetric matrices
Preconditioning
Preconditioning by SSOR
Preconditioning by ILU
Remarks on parallelization
Nonlinear Problems
The Uzawa algorithm and its variants
An alternative
Multigrid Methods / Chapter V:
Multigrid Methods for Variational Problems
Smoothing properties of classical iterative methods
The multigrid idea
The algorithm
Transfer between grids
Convergence of Multigrid Methods
Discrete norms
Connection with the Sobolev norm
Approximation property
Convergence proof for the two-grid method
An alternative short proof
Some variants
Convergence for Several Levels
A recurrence formula for the W-cycle
An improvement for the energy norm
The convergence proof for the V-cycle
Nested Iteration
Computation of starting values
Complexity
Multigrid methods with a small number of levels
The CASCADE algorithm
Multigrid Analysis via Space Decomposition
Schwarz alternating method
Assumptions
Direct consequences
Convergence of multiplicative methods
Verification of A1
Local mesh refinements
The multigrid-Newton method
The nonlinear multigrid method
Starting values
Finite Elements in Solid Mechanics / Chapter VI:
Introduction to Elasticity Theory
Kinematics
The equilibrium equations
The Piola transform
Constitutive Equations
Linear material laws
Hyperelastic Materials
Linear Elasticity Theory
The variational problem
The displacement formulation
The mixed method of Hellinger and Reissner
The mixed method of Hu and Washizu
Nearly incompressible material
Locking
Locking of the Timoshenko beam and typical remedies
Membranes
Plane stress states
Plane strain states
Membrane elements
The PEERS element
Beams and Plates: The Kirchhoff Plate
The hypotheses
Note on beam models
Mixed methods for the Kirchoff plate
DKT elements
The Mindlin-Reissner Plate
The Helmholtz decomposition
The mixed formulation with the Helmholtz decomposition
MITC elements
The Model without a Helmboltz decomposition
References
Index
Preface to the Third English Edition page
Preface to the First English Edition
Preface to the German Edition
4.

図書

図書
Larry L. Schumaker
出版情報: New York : Wiley, c1981  xiv, 553 p. ; 24 cm
シリーズ名: Pure and applied mathematics : a Wiley-Interscience series of texts, monographs, and tracts
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5.

図書

図書
edited by D. Braess, L.L. Schumaker
出版情報: Basel ; Boston : Birkhäuser, 1992-  v. ; 24 cm
シリーズ名: International series of numerical mathematics ; v. 105
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6.

図書

図書
edited by Charles K. Chui, L.L. Schumaker, and J.D. Ward
出版情報: Boston ; Tokyo : Academic Press, c1989  2 v. ; 24 cm
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7.

図書

図書
editors, Gregory E. Fasshauer, Larry L. Schumaker
出版情報: Cham : Springer, c2014  xiii, 395 p. ; 24 cm
シリーズ名: Springer proceedings in mathematics & statistics ; v. 83
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8.

図書

図書
Gregory E. Fasshauer, Larry L. Schumaker, editors
出版情報: [Cham] : Springer, c2017  x, 398 p. ; 24 cm
シリーズ名: Springer proceedings in mathematics & statistics ; v. 201
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9.

図書

図書
edited by E.W. Cheney, C.K. Chui, and L.L. Schumaker
出版情報: Boston : Academic Press, c1993  xix, 249 p. ; 24 cm
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10.

図書

図書
Dietrich Braess ; translated by Larry L. Schumaker
出版情報: Cambridge : Cambridge University Press, 2001  xvii, 352 p. ; 23 cm
所蔵情報: loading…
目次情報: 続きを見る
Preface to the Third English Edition page
Preface to the First English Edition
Preface to the German Edition
Notation
Introduction / Chapter I:
Examples and Classification of PDE's / 1:
Examples
Classification of PDE's
Well-posed Problems
Problems
The Maximum Principle / 2:
Corollaries
Problem
Finite Difference Methods / 3:
Discretization
Discrete maximum principle
A Convergence Theory for Difference Methods / 4:
Consistency
Local and global error
Limits of the convergence theory
Conforming Finite Elements / Chapter II:
Sobolev Spaces
Introduction to Sobolev spaces
Friedrichs' inequality
Possible singularities of H1 functions
Compact imbeddings
Variational Formulation of Elliptic Boundary-Value Problems of Second Order
Variational formulation
Reduction to homogeneous boundary conditions
Existence of solutions
Inhomogeneous boundary conditions
The Neumann Boundary-Value Problem. A Trace Theorem
Ellipticity in H1
Boundary-value problems with natural boundary conditions
Neumann boundary conditions
Mixed boundary conditions
Proof of the trace theorem
Practical consequences of the trace theorem
The Ritz-Galerkin Method and Some Finite Elements
Model Problem
Some Standard Finite Elements / 5:
Requirements on the meshes
Significance of the differentiability properties
Triangular elements with complete polynomials
Remarks on C1 elements
Bilinear elements
Quadratic rectangular elements
Affine families
Choice of an element
Approximation Properties / 6:
The Bramble-Hilbert lemma
Bilinear quadrilateral elements
Inverse estimates
Clément's interpolation
Appendix: On the optimality of the estimates
Error Bounds for Elliptic Problems of Second Order / 7:
Remarks on regularity
Error bounds in the energy norm
L2 estimates
A simple L? estimate
The L2-projector
Computational Considerations / 8:
Assembling the stiffness matrix
Static condensation
Complexity of setting up the matrix
Effect on the choice of a grid
Local mesh refinement
Implementation of the Neumann boundary-value Problem
Nonconforming and Other Methods / Chapter III:
Abstract Lemmas and a Simple Boundary Approximation
Generalizations of Céa's lemma
Duality methods
The Crouzeix-Raviart element
A Simple approximation to curved boundaries
Modifications of the duality argument
Isoparametric Elements
Isoparametric triangular elements
Isoparametric quadrilateral elements
Further Tools from Functional Analysis
Negative norms
Adjoint operators
An abstract existence theorem
An abstract convergence theorem
Proof of Theorem 3.4
Saddle Point Problems
Saddle points and minima
The inf-sup condition
Mixed finite element methods
Fortin interpolation
Saddle point problems with penalty term
Typical applications
Mixed Methods for the Poisson Equation
The Poisson equation as a mixed problem
The Raviart - Thomas element
Interpolation by Raviart-Thomas elements
Implementation and postprocessing
Mesh-dependent norms for the Raviart-Thomas element
The Softening behaviour of mixed methods
The Stokes Equation
Nearly incompressible flows
Finite Elements for the Stokes Problems
An instable element
The Taylor-Hood element
The MINI element
The divergence-free nonconforming P1 element
A Posteriori Error Estimates
Residual estimators
Lower estimates
Remark on other estimators
Local mesh refinement and convergence
A Posteriori Error Estimates via the Hypercircle Method / 9:
The Conjugate Gradient Method / Chapter IV:
Classical Iterative Methods for Solving Linear Systems
Stationary linear processes
The Jacobi and Gauss-Seidel methods
The model problem
Overrelaxation
Gradient Methods
The general gradient method
Gradient methods and quadratic functions
Convergence behavior in the case of large condition numbers
Conjugate Gradient and the Minimal Residual Method
The CG algorithm
Analysis of the CG method as an optimal method
The minimal residual method
Indefinite and unsymmetric matrices
Preconditioning
Preconditioning by SSOR
Preconditioning by ILU
Remarks on parallelization
Nonlinear Problems
The Uzawa algorithm and its variants
An alternative
Multigrid Methods / Chapter V:
Multigrid Methods for Variational Problems
Smoothing properties of classical iterative methods
The multigrid idea
The algorithm
Transfer between grids
Convergence of Multigrid Methods
Discrete norms
Connection with the Sobolev norm
Approximation property
Convergence proof for the two-grid method
An alternative short proof
Some variants
Convergence for Several Levels
A recurrence formula for the W-cycle
An improvement for the energy norm
The convergence proof for the V-cycle
Nested Iteration
Computation of starting values
Complexity
Multigrid methods with a small number of levels
The CASCADE algorithm
Multigrid Analysis via Space Decomposition
Schwarz alternating method
Assumptions
Direct consequences
Convergence of multiplicative methods
Verification of A1
Local mesh refinements
The multigrid-Newton method
The nonlinear multigrid method
Starting values
Finite Elements in Solid Mechanics / Chapter VI:
Introduction to Elasticity Theory
Kinematics
The equilibrium equations
The Piola transform
Constitutive Equations
Linear material laws
Hyperelastic Materials
Linear Elasticity Theory
The variational problem
The displacement formulation
The mixed method of Hellinger and Reissner
The mixed method of Hu and Washizu
Nearly incompressible material
Locking
Locking of the Timoshenko beam and typical remedies
Membranes
Plane stress states
Plane strain states
Membrane elements
The PEERS element
Beams and Plates: The Kirchhoff Plate
The hypotheses
Note on beam models
Mixed methods for the Kirchoff plate
DKT elements
The Mindlin-Reissner Plate
The Helmholtz decomposition
The mixed formulation with the Helmholtz decomposition
MITC elements
The Model without a Helmboltz decomposition
References
Index
Preface to the Third English Edition page
Preface to the First English Edition
Preface to the German Edition
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