Elliptic differential equations : theory and numerical treatment
 Responsibility
 Wolfgang Hackbusch.
 Edition
 Second edition.
 Publication
 [Berlin] : Springer, [2017]
 Physical description
 xiv, 455 pages ; 24 cm.
 Series
 Springer series in computational mathematics ; 18.
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA372 .H1213 2017  Unknown 
More options
Description
Creators/Contributors
 Author/Creator
 Hackbusch, W., 1948 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 429441) and index.
 Contents

 Partial differential equations and their classification into types
 Examples
 Classification of secondorder equations into types
 Type classification for systems of first order
 Characteristic properties of the different types
 Literature
 The potential equation
 Posing the problem
 Singularity function
 Meanvalue property and maximum principle
 Continuous dependence on the boundary data
 The poisson equation
 Posing the problem
 Representation of the solution by the green function
 Existence of a solution
 The green gunction for the ball
 The neumann boundaryvalue problem
 The integral equation method
 Difference methods for the poisson equation
 Introduction : the onedimensional case
 The fivepoint formula
 Mmatrices, matrix norms, positivedefinite matrices
 Properties of the matrix Lh
 Convergence
 Discretisations of higher order
 The discretisation of the neumann boundaryvalue problem
 Onesided difference for ...
 Symmetric difference for ...
 Symmetric difference for ... on an offset grid
 Proof of the stability theorem 4.62
 Discretisation in an arbitrary domain
 Shortleyweller approximation
 Interpolation in nearboundary points
 General boundaryvalue problems
 Dirichlet boundaryvalue problems for linear differential equations
 Posing the problem
 Maximum principle
 Uniqueness of the solution and continuous dependence
 Difference methods for the general differential equation of second order
 Green's function
 General boundary conditions
 Formulating the boundaryvalue problem
 Difference methods for general boundary conditions
 Boundary problems of higher order
 The biharmonic differential equation
 General linear differential equations of order 2m
 Discretisation of the biharmonic differential equation
 Tools from functional analysis
 Banach spaces and hilbert spaces
 Normed spaces
 Operators
 Banach spaces
 Hilbert spaces
 Sobolev spaces
 L²(...)
 Hk(...) and Hk(...)
 Fourier transformation and Hk (Rn)
 Hs(...) for Real s ...
 Trace and extension theorems
 Dual spaces
 Dual space of a normed space
 Adjoint operators
 Scales of hilbert spaces
 Compact operators
 Bilinear forms
 Variational formulation
 Historical remarks about the dirichlet principle
 Equations with homogeneous dirichlet boundary conditions
 Dirichlet boundary condition
 Weak formulation
 ... (...)Ellipticity
 ... (...)Coercivity
 Inhomogeneous dirichlet boundary conditions
 Natural boundary conditions
 Variation in Hm(...)
 Conormal boundary condition
 Oblique boundary condition
 Boundary conditions for m ... 2
 Further boundary conditions
 Pseudodifferential equations
 The finiteelement method
 Historical remarks
 The ritzgalerkin method
 Basics
 Analysis of the discrete equation
 Solvability of the discrete problem
 Examples
 Error estimates
 Quasioptimality
 Convergence of the ritzgalerkin solutions
 Ritz projection
 Further stability and error estimates
 Finite elements
 Introduction : linear elements for ... = (a, b)
 Linear Elements for ... R²
 Bilinear Elements for ... R²
 Quadratic Elements for ... K²
 Elements for ... U³
 Handling of side conditions
 Error estimates for finiteelement methods
 Preparations
 Properties of sequences of finiteelement spaces
 H¹Estimates for linear elements
 L² Estimates for linear elements
 Generalisations
 Error estimates for other elements
 Finite elements for equations of higher order
 Finite elements for nonpolygonal regions
 Aposteriori error estimates, adaptivity
 Aposteriori error estimates
 Efficiency of the finiteelement method
  Properties of the system matrix
 Connection of L and Lh
 Equivalent norms and mass matrix
 Inverse estimate and condition of L
 Element matrices
 Positivity, Maximum principle
 Further remarks
 Mixed and hybrid finiteelement methods
 Nonconforming elements
 Inadmissible triangulations
 Trefftz' method
 Finiteelement methods for singular solutions
 Hierarchical bases
 Superconvergence
 Mortar finite elements
 Composite finite elements
 Related discretisations
 Sparse grids
 Regularity
 Solutions of the boundaryvalue problem in Hs(...). s > m
 The regularity problem
 Regularity theorems for ... = ... R"
 Regularity theorems for ... = Rn
 Regularity theorems for general domains ...
 Regularity for convex domains and domains with corners
 Regularity in the interior
 Estimates
 Behaviour of the singularity and green's function
 Regularity properties of difference equations
 Discrete H¹regularity
 Consistency
 Optimal error estimates
 H...regularity for 1/2 <...<l/2
 H ...regularity
 Interior regularity
 Special differential equations
 Differential equations with discontinuous coefficients
 Formulation
 Finiteelement discretisation
 Discretisation by difference schemes
 Discontinuous coefficients of the first and zeroth derivatives
 A singular perturbation problem
 The convectiondiffusion equation
 Stable difference schemes
 Finite elements
 Elliptic eigenvalue problems
 Formulation of eigenvalue problems
 Finiteelement discretisation
 Discretisation
 Qualitative convergence results
 Quantitative convergence results
 Consistent problems
 Discretisation by difference methods
 Further remarks
 Stokes equations
 Elliptic systems of differential equations
 Variational formulation
 Weak formulation of the stokes equations
 Saddlepoint problems
 Existence and uniqueness of the solution of a saddlepoint problem
 Solvability and regularity of the stokes problem
 A V0elliptic variational formulation of the stokes problem
 Finiteelement method for the stokes problem
 Finiteelement discretisation of a saddlepoint problem
 Stability conditions
 Stable finiteelement spaces for the stokes problem
 Divergencefree elements
 Solution of the exercises
 Exercises of chapter 1
 Exercises of chapter 2
 Exercises of chapter 3
 Exercises of chapter 4
 Exercises of chapter 5
 Exercises of chapter 6
 Exercises of chapter 7
 Exercises of chapter 8
 Exercises of chapter 9
 Exercises of chapter 10
 Exercises of chapter 11
 Exercises of chapter 12
 References
 List of symbols and abbreviations
 Index.
 Publisher's Summary
 ["This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finiteelement methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddlepoint problem taking into account its relevance to applications in fluid dynamics.", {"source"=>"(source: Nielsen Book Data)"}, "9783662549605", "20170911"]
Subjects
Bibliographic information
 Publication date
 2017
 Series
 Springer series in computational mathematics, 01793632 ; 18
 ISBN
 9783662549605 (English edition : hd.bd.)
 9783662549612 (online)
 3662549603 hardcover