Analysis of finite difference schemes : for linear partial differential equations with generalized solutions
 Author/Creator
 Jovanović, Boško S., author.
 Language
 English.
 Publication
 London ; New York : Springer, [2014]
 Physical description
 xiii, 408 pages : illustrations ; 25 cm.
 Series
 Springer series in computational mathematics ; 46.
Access
Available online

Stacks

Unknown
QA377 .J684 2014

Unknown
QA377 .J684 2014
More options
Contributors
 Contributor
 Süli, Endre, 1956 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 389398) and index.
 Contents

 Distributions and function spaces
 Elements of functional analysis
 A survey of abstract spaces
 Linear operators in normed linear spaces
 Sublinear functionals
 Linear functionals on hilbert spaces
 Interpolation of banach spaces
 Basic function spaces
 Spaces of continuous functions
 Spaces of integrable functions
 Distributions
 Test functions and distributions
 Operations with distributions
 Tempered distributions
 Fourier transform of a tempered distribution
 Sobolev spaces
 Anisotropie sobolev spaces
 Besov spaces
 Interpolation properties of sobolev spaces
 Multiplier spaces
 Fourier multipliers and mollifiers
 Fourier multipliers
 Definition of mollifier
 An admissible class of mollifiers
 Mollifiers of tempered distributions
 Multipliers and mollifiers on periodic spaces
 Elliptic boundaryvalue problems
 Existence and uniqueness of solutions
 Approximation of elliptic problems
 Introduction to the theory of finite difference schemes
 Finite difference approximation in one space dimension
 Finite difference scheme for a univariate problem
 The multidimensional case
 Approximation of a generalized poisson problem
 Convergence analysis on uniform meshes
 The BrambleHilbert Lemma
 Optimal error bounds on uniform meshes
 Convergence analysis on nonuniform meshes
 Cartesianproduct nonuniform meshes
 An alternative scheme
 The rotated discrete laplacian
 Convergence analysis in Lp norms
 Discrete fourier multipliers
 The model problem and its approximation
 Convergence in discrete besselpotential norms
 Approximation of secondorder elliptic equations with variable coefficients
 Convergence in the discrete W¹2 norm
 Convergence in the discreteW²2 norm
 Convergence in the discrete L2 norm
 Convergence in discrete fractionalorder norms
 Convergence in the discrete L2 norm : separated variables
 Fourthorder elliptic equations
 An elliptic interface problem
 Finite difference approximation
 Convergence in the discrete W¹2 norm
 Bibliographical notes
 Finite difference approximation of parabolic problems
 Parabolicequations
 Abstract parabolic initialvalue problems
 Some a priori estimates
 Application to parabolic partial differential equations
 Abstract twolevel operatordifference schemes
 Classical difference schemes for the heat equation
 Explicit and implicit schemes
 Stability of explicit and implicit schemes
 Error analysis of difference schemes for the heat equation
 The heatequation with nonsmooth data
 The initialboundaryvalue problem and its discretization
 Error analysis
 The case of independent meshsizes
 Parabolic problems with variable coefficients
 Formulation of the problem
 The finite difference scheme
 Error analysis
 Factorized scheme
 A parabolic interface problem
 Finite difference approximation
 Factorized scheme
 A parabolic transmission problem
 Weak solutions and function spaces
 Finite difference approximation
 Factorized scheme
 Bibliographical notes
 Finite difference approximation of hyperbolic problems
 Hyperbolic equations
 Abstract hyperbolic initialvalue problems
 Some a priori estimates
 Abstract threelevel operatordifference schemes
 Classical difference schemes for the wave equation
 Explicit and weighted schemes
 Stability of the weighted difference scheme
 Error analysis of difference schemes for the wave equation
 The wave equation with nonsmooth data
 The initialboundaryvalue problem and its discretization
 Error analysis
 Hyperbolic problems with variable coefficients
 Formulation of the problem
 The finite difference scheme
 Convergence of the finite difference scheme
 Factorized scheme
 Hyperbolic interface problem
 Finite difference approximation
 Factorized scheme
 Hyperbolic transmission problem
 Finite difference approximation
 Bibliographical notes
 References
 Index
 List of symbols.
 Publisher's Summary
 This book develops a systematic and rigorous mathematical theory of finite difference methods for linear elliptic, parabolic and hyperbolic partial differential equations with nonsmooth solutions. Finite difference methods are a classical class of techniques for the numerical approximation of partial differential equations. Traditionally, their convergence analysis presupposes the smoothness of the coefficients, source terms, initial and boundary data, and of the associated solution to the differential equation. This then enables the application of elementary analytical tools to explore their stability and accuracy. The assumptions on the smoothness of the data and of the associated analytical solution are however frequently unrealistic. There is a wealth of boundary  and initial  value problems, arising from various applications in physics and engineering, where the data and the corresponding solution exhibit lack of regularity. In such instances classical techniques for the error analysis of finite difference schemes break down. The objective of this book is to develop the mathematical theory of finite difference schemes for linear partial differential equations with nonsmooth solutions. Analysis of Finite Difference Schemes is aimed at researchers and graduate students interested in the mathematical theory of numerical methods for the approximate solution of partial differential equations.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2014
 Responsibility
 Boško S. Jovanović, Endre Süli.
 Title Variation
 For linear partial differential equations with generalized solutions
 Series
 Springer series in computational mathematics, 01793632 ; 46
 Available in another form
 9781447154600 (online)
 (GyWOH)har135025875
 ISBN
 9781447154594
 1447154592