Numerical Analysis: Historical Developments in the 20th Century

C. Brezinski, L. Wuytack

• Publisher: Elsevier Science
• Year: 2012
• Print ISBN: 9780444506177
• eBook ISBN: 9780444598585
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Numerical analysis has witnessed many significant developments in the 20th century. This book brings together 16 papers dealing with historical developments, survey papers and papers on recent trends in selected areas of numerical analysis, such as: approximation and interpolation, solution of linear systems and eigenvalue problems, iterative methods, quadrature rules, solution of ordinary-, partial- and integral equations. The papers are reprinted from the 7-volume project of the Journal of Computational and Applied Mathematics on '/homepage/sac/cam/na2000/index.htmlNumerical Analysis 2000'. An introductory survey paper deals with the history of the first courses on numerical analysis in several countries and with the landmarks in the development of important algorithms and concepts in the field.

From finite differences to finite elements A short history of numerical analysis of partial differential equations

Vidar Thomée [email protected] Department of Mathematics, Chalmers University of Technology,S-4I2 96 Göteborg, Sweden

Abstract

This is an account of the history of numerical analysis of partial differential equations, starting with the 1928 paper of Courant, Friedrichs, and Lewy, and proceeding with the development of first finite difference and then finite element methods. The emphasis is on mathematical aspects such as stability and convergence analysis. © 2001 Elsevier Science B.V. All rights reserved.

MSC: 01A60; 65-03; 65M10; 65N10; 65M60; 65N30

Keywords

History

Finite difference methods

Finite element methods

0 Introduction

This article is an attempt to give a personal account of the development of numerical analysis of partial differential equations. We begin with the introduction in the 1930s and further development of the finite difference method and then describe the subsequent appearence around 1960 and increasing role of the finite element method. Even though clearly some ideas may be traced back further, our starting point will be the fundamental theoretical paper by Courant, Friedrichs and Lewy (1928)¹ on the solution of problems of mathematical physics by means of finite differences. In this paper a discrete analogue of Dirichlet’s principle was used to define an approximate solution by means of the five-point approximation of Laplace’s equation, and convergence as the mesh width tends to zero was established by compactness. A finite difference approximation ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• Numerical analysis in the twentieth century
• Approximation in normed linear spaces
• A tutorial history of least squares with applications to astronomy and geodesy
• Convergence acceleration during the 20th century
• On the history of multivariate polynomial interpolation
• Numerical linear algebra algorithms and software
• Iterative solution of linear systems in the 20th century
• Eigenvalue computation in the 20th century
• Historical developments in convergence analysis for Newton’s and Newton-like methods
• A survey of truncated-Newton methods
• Cubature formulae and orthogonal polynomials
• Computation of Gauss-type quadrature formulas
• A review of algebraic multigrid
• From finite differences to finite elements A short history of numerical analysis of partial differential equations
• A perspective on the numerical treatment of Volterra equations
• Numerical methods for ordinary differential equations in the 20th century
• Retarded differential equations