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2009 | Buch

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Theoretical Numerical Analysis

A Functional Analysis Framework

verfasst von: Weimin Han, Kendall E. Atkinson

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

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

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Über dieses Buch

This textbook prepares graduate students for research in numerical analysis/computational mathematics by giving to them a mathematical framework embedded in functional analysis and focused on numerical analysis. This helps the student to move rapidly into a research program. The text covers basic results of functional analysis, approximation theory, Fourier analysis and wavelets, iteration methods for nonlinear equations, finite difference methods, Sobolev spaces and weak formulations of boundary value problems, finite element methods, elliptic variational inequalities and their numerical solution, numerical methods for solving integral equations of the second kind, and boundary integral equations for planar regions. The presentation of each topic is meant to be an introduction with certain degree of depth. Comprehensive references on a particular topic are listed at the end of each chapter for further reading and study.

Because of the relevance in solving real world problems, multivariable polynomials are playing an ever more important role in research and applications. In this third editon, a new chapter on this topic has been included and some major changes are made on two chapters from the previous edition. In addition, there are numerous minor changes throughout the entire text and new exercises are added.

Review of earlier edition:

"...the book is clearly written, quite pleasant to read, and contains a lot of important material; and the authors have done an excellent job at balancing theoretical developments, interesting examples and exercises, numerical experiments, and bibliographical references."

R. Glowinski, SIAM Review, 2003

Inhaltsverzeichnis

Frontmatter

1. Linear Spaces

Abstract

Linear (or vector) spaces are the standard setting for studying and solving a large proportion of the problems in differential and integral equations,approximation theory, optimization theory, and other topics in applied mathematics. In this chapter, we gather together some concepts and results concerning various aspects of linear spaces, especially some of the more important linear spaces such as Banach spaces, Hilbert spaces, and certain function spaces that are used frequently in this work and in applied mathematics generally.

Kendall Atkinson, Weimin Han

2. Linear Operators on Normed Spaces

Abstract

Many of the basic problems of applied mathematics share the property of linearity, and linear spaces and linear operators provide a general and useful framework for the analysis of such problems.More complicated applications often involve nonlinear operators, and a study of linear operators also offers some useful tools for the analysis of nonlinear operators. In this chapter we review some basic results on linear operators, and we give some illustrative applications to obtain results in numerical analysis. Some of the results are quoted without proof; and usually the reader can find detailed proofs of the results in a standard textbook on functional analysis, e.g. see Conway [58], Kantorovich and Akilov [135], and Zeidler [249], [250].

Kendall Atkinson, Weimin Han

3. Approximation Theory

Abstract

In this chapter, we deal with the problem of approximation of functions. A prototype problem can be described as follows: For some function f, known exactly or approximately, find an approximation that has a more simply computable form, with the error of the approximation within a given error tolerance. Often the function f is not known exactly. For example, if the function comes from a physical experiment, we usually have a table of function values only. Even when a closed-form expression is available, it may happen that the expression is not easily computable, for example,

Kendall Atkinson, Weimin Han

4. Fourier Analysis and Wavelets

Abstract

In this chapter, we provide an introduction to the theory of Fourier analysis and wavelets. Fourier analysis is a large branch of mathematics, and it is useful in a wide spectrum of applications, such as in solving differential equations arising in sciences and engineering, and in signal processing. The first three sections of the chapter will be devoted to the Fourier series, the Fourier transform, and the discrete Fourier transform, respectively. The Fourier transform converts a function of a time or space variable into a function of a frequency variable. When the original function is periodic, it is sufficient to consider integer multiples of the base frequency, and we are led to the notion of the Fourier series. For a general non-periodic function, we need coefficients of all possible frequencies, and the result is the Fourier transform.

Kendall Atkinson, Weimin Han

5. Nonlinear Equations and Their Solution by Iteration

Abstract

Nonlinear functional analysis is the study of operators lacking the property of linearity. In this chapter, we consider nonlinear operator equations and their numerical solution. We begin the consideration of operator equations which take the form

Kendall Atkinson, Weimin Han

6. Finite Difference Method

Abstract

The finite difference method is a universally applicable numerical method for the solution of differential equations. In this chapter, for a sample parabolic partial differential equation, we introduce some difference schemes and analyze their convergence. We present the well-known Lax equivalence theorem and related theoretical results, and apply them to the convergence analysis of difference schemes.

The finite difference method can be difficult to analyze, in part because it is quite general in its applicability. Much of the existing stability and convergence analysis is restricted to special cases, particularly to linear differential equations with constant coefficients. These results are then used to predict the behavior of difference methods for more complicated equations.

Kendall Atkinson, Weimin Han

7. Sobolev Spaces

Abstract

In this chapter, we review definitions and properties of Sobolev spaces, which are indispensable for a theoretical analysis of partial differential equations and boundary integral equations, as well as being necessary for the analysis of some numerical methods for solving such equations. Most results are stated without proof; proofs of the results and detailed discussions of Sobolev spaces can be found in numerous monographs and textbooks, e.g. [1, 78, 89, 245].

Kendall Atkinson, Weimin Han

8. Weak Formulations of Elliptic Boundary Value Problems

Abstract

In this chapter, we consider weak formulations of some elliptic boundary value problems and study the well-posedness of the variational problems. We begin with a derivation of a weak formulation of the homogeneous Dirichlet boundary value problem for the Poisson equation. In the abstract form, a weak formulation can be viewed as an operator equation. In the second section, we provide some general results on existence and uniqueness for linear operator equations. In the third section, we present and discuss the well-known Lax-Milgram Lemma, which is applied, in the section following, to the study of well-posedness of variational formulations for various linear elliptic boundary value problems. We also apply the Lax-Milgram Lemma in studying a boundary value problem in linearized elasticity; this is done in Section 8.5. The framework in the Lax-Milgram Lemma is suitable for the development of the Galerkin method for numerically solving linear elliptic boundary value problems. In Section 8.6, we provide a brief discussion of two different weak formulations: the mixed formulation and the dual formulation. For the development of Petrov-Galerkin method, where the trial function space and the test function space are different, we discuss a generalization of Lax-Milgram Lemma in Section 8.7. Most of the chapter is concerned with boundary value problems with linear differential operators. In the last section, we analyze a nonlinear elliptic boundary value problem.

Kendall Atkinson, Weimin Han

9. The Galerkin Method and Its Variants

Abstract

In this chapter, we briefly discuss some numerical methods for solving boundary value problems. These are the Galerkin method and its variants: the Petrov-Galerkin method and the generalized Galerkin method. In Section 9.4, we rephrase the conjugate gradient method, discussed in Section 5.6, for solving variational equations.

The Galerkin method provides a general framework for approximation of operator equations, which includes the finite element method as a special case. In this section, we discuss the Galerkin method for a linear operator equation in a form directly applicable to the study of the finite element method.

Kendall Atkinson, Weimin Han

10. Finite Element Analysis

Abstract

The finite element method is the most popular numerical method for solving elliptic boundary value problems. In this chapter, we introduce the concept of the finite element method, the finite element interpolation theory and its application in error estimates of finite element solutions of elliptic boundary value problems. The boundary value problems considered in this chapter are linear.

From the discussion in the previous chapter, we see that the Galerkin method for a linear boundary value problem reduces to the solution of a linear system. In solving the linear system, properties of the coefficient matrix A play an essential role. For example, if the condition number of A is too big, then from a practical perspective, it is impossible to find directly an accurate solution of the system (see [15]). Another important issue is the sparsity of the matrix A. The matrix A is said to be sparse, if most of its entries are zero; otherwise the matrix is said to be dense. Sparseness of the matrix can be utilized for two purposes. First, the stiffness matrix is less costly to form (observing that the computation of each entry of the matrix involves a domain integration and sometimes a boundary integration as well). Second, if the coefficient matrix is sparse, then the linear system can usually be solved more efficiently. To get a sparse stiffness matrix with the Galerkin method, we use finite dimensional approximation spaces such that it is possible to choose basis functions with small support. This consideration gives rise to the idea of the finite element method, where we use piecewise (images of) smooth functions (usually polynomials) for approximations. Loosely speaking, the finite element method is a Galerkin method with the use of piecewise (images of) polynomials.

Kendall Atkinson, Weimin Han

11. Elliptic Variational Inequalities and Their Numerical Approximations

Abstract

Variational inequalities form an important family of nonlinear problems. Some of the more complex physical processes are described by variational inequalities. We study some elliptic variational inequalities (EVIs) in this chapter, presenting results on existence, uniqueness and stability of solutions to the EVIs, and discussing their numerical approximations.

We start with an introduction of two sample elliptic variational inequalities (EVIs) in Section 11.1. Many elliptic variational inequalities arising in mechanics can be equivalently expressed as convex minimization problems; a study of such variational inequalities is given in Section 11.2. Then in Section 11.3, we consider a general family of EVIs that do not necessarily relate to minimization problems. Numerical approximations of the EVIs are the topics of Section 11.4. In the last section, we consider some contact problems in elasticity that lead to EVIs.

Kendall Atkinson, Weimin Han

12. Numerical Solution of Fredholm Integral Equations of the Second Kind

Abstract

Linear integral equations of the second kind, were introduced in Chapter 2, and we note that they occur in a wide variety of physical applications. An important class of such equations are the boundary integral equations, about which more is said in Chapter 13. In the integral of (12.0.1), D is a closed, and often bounded, integration region. The integral operator is often a compact operator on C(D) or L2(D), although not always. For the case that the integral operator is compact, a general solvability theory is given in Subsection 2.8.4 of Chapter 2. A more general introduction to the theory of such equations is given in Kress [149].

Kendall Atkinson, Weimin Han

13. Boundary Integral Equations

Abstract

In Chapter 10, we examined finite element methods for the numerical solution of Laplace's equation. In this chapter, we propose an alternative approach. We introduce the idea of reformulating Laplace's equation as a boundary integral equation (BIE), and then we consider the numerical solution of Laplace's equation by numerically solving its reformulation as a BIE. Some of the most important boundary value problems for elliptic partial differential equations have been studied and solved numerically by this means; and depending on the requirements of the problem, the use of BIE reformulations may be the most efficient means of solving these problems. Examples of other equations solved by use of BIE reformulations are the Helmholtz equation (Δυ + λυ = 0) and the biharmonic equation (Δ²υ = 0). We consider here the use of boundary integral equations in solving only planar problems for Laplace's equation. For the domain D for the equation, we restrict it or its complement to be a simply-connected set with a smooth boundary S. Most of the results and methods given here will generalize to other equations (e.g. Helmholtz's equation).

Kendall Atkinson, Weimin Han

14. Multivariable Polynomial Approximations

Abstract

In Chapter 3 we introduced the approximation of univariate functions by polynomials and trigonometric functions (see Sections 3.4–3.7). In this chapter we extend those ideas to multivariable functions and multivariable polynomials. In the univariate case there are only three types of approximation domains, namely [a, b], [a,∞), and (–∞,∞), with a and b finite. In contrast, there are many types of approximation domains in the multivariable case. This chapter is only an introduction to this area, and to make it more accessible, we emphasize planar problems. In particular, we consider the unit disk

Kendall Atkinson, Weimin Han

Backmatter

Titel: Theoretical Numerical Analysis
verfasst von: Weimin Han
Kendall E. Atkinson
Verlag: Springer New York
Electronic ISBN: 978-1-4419-0458-4
Print ISBN: 978-1-4419-0457-7
DOI: https://doi.org/10.1007/978-1-4419-0458-4