Analysis of Finite Difference Schemes

For Linear Partial Differential Equations with Generalized Solutions

verfasst von: Boško S. Jovanović, Endre Süli

Verlag: Springer London

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Distributions and Function Spaces

Abstract

Chapter 1 provides a brief survey of basic results from linear functional analysis, particularly Banach and Hilbert space theory, and an overview of standard results from the theory of distributions and function spaces, including isotropic and anisotropic Sobolev spaces, Besov spaces, Fourier multipliers and mollifiers in function spaces, and function space interpolation.

Boško S. Jovanović, Endre Süli

Chapter 2. Elliptic Boundary-Value Problems

Abstract

Chapter 2 is concerned with the construction and the convergence analysis of finite difference schemes for elliptic boundary-value problems. One of the key contributions of the chapter is the derivation of optimal-order bounds on the error between the analytical solution and its finite difference approximation for elliptic equations with variable coefficients under minimal regularity hypotheses on the coefficients and the solution, the minimal regularity hypotheses on the coefficients being expressed in terms of spaces of multipliers in Sobolev spaces. The construction of the difference schemes is based on the use of mollifiers and a key tool in the convergence analysis of the schemes are the Bramble–Hilbert lemma and its various generalizations.

Boško S. Jovanović, Endre Süli

Chapter 3. Finite Difference Approximation of Parabolic Problems

Abstract

Chapter 3 is concerned with the construction and the convergence analysis of finite difference schemes for parabolic initial-boundary-value problems. The central contribution of the chapter is the derivation of optimal-order bounds on the error between the analytical solution and its finite difference approximation for parabolic equations with variable coefficients under minimal regularity hypotheses on the coefficients and the solution, the minimal regularity hypotheses on the coefficients being expressed in terms of spaces of multipliers in anisotropic Sobolev spaces.

Boško S. Jovanović, Endre Süli

Chapter 4. Finite Difference Approximation of Hyperbolic Problems

Abstract

Chapter 4 is concerned with the construction and the convergence analysis of finite difference schemes for hyperbolic initial-boundary-value problems. A key contribution of the chapter is the derivation of optimal-order bounds on the error between the analytical solution and its finite difference approximation for hyperbolic equations with variable coefficients under minimal regularity hypotheses on the coefficients and the solution, the minimal regularity hypotheses on the coefficients being expressed in terms of spaces of multipliers in anisotropic Sobolev spaces.

Boško S. Jovanović, Endre Süli

Backmatter

Titel: Analysis of Finite Difference Schemes
verfasst von: Boško S. Jovanović
Endre Süli
Verlag: Springer London
Electronic ISBN: 978-1-4471-5460-0
Print ISBN: 978-1-4471-5459-4
DOI: https://doi.org/10.1007/978-1-4471-5460-0