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To Anita Floris Ingrid and Patty,Preface vii,Part I Introduction 1. 1 The Vision of Leibniz 3,2 A Brief History 11,2 1 A timeline for calculus 12. 2 2 A timeline for the computer 15,2 3 An important mathematical controversy 17. 2 4 Some important personalities 19,3 A Review of Calculus 25. 3 1 Real numbers Sequences of real numbers 26,3 2 Functions 29.

3 3 Differential equations 38,3 4 The Fundamental Theorem of Calculus 41. 3 5 A look ahead 58,4 A Short Review of Linear Algebra 61. 4 1 Linear combinations linear independency and basis 62. 4 2 Norms inner products and orthogonality 63,4 3 Linear transformations and matrices 67. 4 4 Eigenvalues and eigenvectors 68,4 5 Norms of matrices 70. 4 6 Vector spaces of functions 71,5 Polynomial Approximation 75.

5 1 Vector spaces of polynomials 75,5 2 Polynomial interpolation 78. vi CONTENTS,5 3 Vector spaces of piecewise polynomials 87. 5 4 Interpolation by piecewise polynomials 90,5 5 Quadrature 95. 5 6 The L2 projection into a space of polynomials 100. 5 7 Approximation by trigonometric polynomials 102. 6 Galerkin s Method 104,6 1 Galerkin s method with global polynomials 105. 6 2 Galerkin s method with piecewise polynomials 113. 6 3 Galerkin s method with trigonometric polynomials 122. 6 4 Comments on Galerkin s method 125,6 5 Important issues 127.

7 Solving Linear Algebraic Systems 129,7 1 Stationary systems of masses and springs 131. 7 2 Direct methods 136,7 3 Direct methods for special systems 146. 7 4 Iterative methods 151,7 5 Estimating the error of the solution 165. Part II The archetypes 171,8 Two Point Boundary Value Problems 173. 8 1 The finite element method 178, 8 2 Error estimates and adaptive error control 189.

8 3 Data and modeling errors 197,8 4 Higher order finite element methods 199. 8 5 The elastic beam 202, 8 6 A comment on a priori and a posteriori analysis 203. 9 Scalar Initial Value Problems 205,9 1 A model in population dynamics 207. 9 2 Galerkin finite element methods 209, 9 3 A posteriori error analysis and adaptive error control 217. 9 4 A priori error analysis 228,9 5 Quadrature errors 233.

9 6 The existence of solutions 237,CONTENTS vii,10 Initial Value Problems for Systems 241. 10 1 Model problems 242, 10 2 The existence of solutions and Duhamel s formula 249. 10 3 Solutions of autonomous problems 251,10 4 Solutions of non autonomous problems 254. 10 5 Stability 255,10 6 Galerkin finite element methods 261. 10 7 Error analysis and adaptive error control 263. 10 8 The motion of a satellite 271,11 Calculus of Variations 275.

11 1 Minimization problems 276,11 2 Hamilton s principle 280. 12 Computational Mathematical Modeling 284,Part III Problems in several dimensions 291. 13 Piecewise Polynomials in Several Dimensions 293. 13 1 Meshes in several dimensions 294,13 2 Vector spaces of piecewise polynomials 297. 13 3 Error estimates for piecewise linear interpolation 307. 13 4 Quadrature in several dimensions 312,14 The Poisson Equation 316. 14 1 The finite element method for the Poisson equation 327. 14 2 Energy norm error estimates 342,14 3 Adaptive error control 346.

14 4 Dealing with different boundary conditions 349. 14 5 Error estimates in the L2 norm 358,15 The Heat Equation 367. 15 1 Maxwell s equations 367, 15 2 The basic structure of solutions of the heat equation 369. 15 3 Stability 376, 15 4 A finite element method for the heat equation 378. 15 5 Error estimates and adaptive error control 382. 15 6 Proofs of the error estimates 385,viii CONTENTS. 16 The Wave Equation 394,16 1 Transport in one dimension 395.

16 2 The wave equation in one dimension 398,16 3 The wave equation in higher dimensions 406. 16 4 A finite element method 411, 16 5 Error estimates and adaptive error control 415. 17 Stationary Convection Diffusion Problems 425,17 1 A basic model 426. 17 2 The stationary convection diffusion problem 428. 17 3 The streamline diffusion method 433,17 4 A framework for an error analysis 437. 17 5 A posteriori error analysis in one dimension 441. 17 6 Error analysis in two dimensions 443,17 7 79 A D 447.

18 Time Dependent Convection Diffusion Problems 448. 18 1 Euler and Lagrange coordinates 450,18 2 The characteristic Galerkin method 454. 18 3 The streamline diffusion method on an Euler mesh 459. 18 4 Error analysis 463, 19 The Eigenvalue Problem for an Elliptic Operator 469. 19 1 Computation of the smallest eigenvalue 473,19 2 On computing larger eigenvalues 474. 19 3 The Schro dinger equation for the hydrogen atom 477. 19 4 The special functions of mathematical physics 480. 20 The Power of Abstraction 482,20 1 The abstract formulation 483. 20 2 The Lax Milgram theorem 485,20 3 The abstract Galerkin method 487.

20 4 Applications 488, 20 5 A strong stability estimate for Poisson s equation 497. I admit that each and every thing remains in its state until there. is reason for change Leibniz, I m sick and tired of this schism between earth and sky. Idealism and realism sorely our reason try Gustaf Fro ding. This book together with the companion volumes Introduction to Computa. tional Differential Equations and Advanced Computational Differential Equa. tions presents a unified approach to computational mathematical modeling. using differential equations based on a principle of a fusion of mathematics and. computation The book is motivated by the rapidly increasing dependence on. numerical methods in mathematical modeling driven by the development of. powerful computers accessible to everyone Our goal is to provide a student. with the essential theoretical and computational tools that make it possible to. use differential equations in mathematical modeling in science and engineering. effectively The backbone of the book is a new unified presentation of numerical. solution techniques for differential equations based on Galerkin methods. Mathematical modeling using differential and integral equations has formed. the basis of science and engineering since the creation of calculus by Leibniz. and Newton Mathematical modeling has two basic dual aspects one symbolic. and the other constructive numerical which reflect the duality between the in. finite and the finite or the continuum and the discrete The two aspects have. been closely intertwined throughout the development of modern science from. the development of calculus in the work of Euler Lagrange Laplace and Gauss. into the work of von Neumann in our time For example Laplace s monumental. Me canique Ce leste in five volumes presents a symbolic calculus for a mathe. matical model of gravitation taking the form of Laplace s equation together. with massive numerical computations giving concrete information concerning. the motion of the planets in our solar system, However beginning with the search for rigor in the foundations of calculus. in the 19th century a split between the symbolic and constructive aspects. gradually developed The split accelerated with the invention of the electronic. computer in the 1940s after which the constructive aspects were pursued in the. new fields of numerical analysis and computing sciences primarily developed. outside departments of mathematics The unfortunate result today is that. symbolic mathematics and constructive numerical mathematics by and large. are separate disciplines and are rarely taught together Typically a student. first meets calculus restricted to its symbolic form and then much later in a. different context is confronted with the computational side This state of affairs. lacks a sound scientific motivation and causes severe difficulties in courses in. physics mechanics and applied sciences building on mathematical modeling. The difficulties are related to the following two basic questions i How to. get applications into mathematics education ii How to use mathematics in. applications Since differential equations are so fundamental in mathematical. modeling these questions may be turned around as follows i How can we. teach differential equations in mathematics education ii How can we use. differential equations in applications, Traditionally the topic of differential equations in basic mathematics edu. cation is restricted to separable scalar first order ordinary differential equations. and constant coefficient linear scalar n th order equations for which explicit so. lution formulas are presented together with some applications of separation of. variables techniques for partial differential equations like the Poisson equation. on a square Even slightly more general problems have to be avoided because. the symbolic solution methods quickly become so complex Unfortunately the. presented tools are not sufficient for applications and as a result the student. must be left with the impression that mathematical modeling based on sym. bolic mathematics is difficult and only seldom really useful Furthermore the. numerical solution of differential equations considered with disdain by many. pure mathematicians is often avoided altogether or left until later classes. where it is often taught in a cookbook style and not as an integral part of. a mathematics education aimed at increasing understanding The net result is. that there seems to be no good answer to the first question in the traditional. mathematics education, The second question is related to the apparent principle of organization of.

a technical university with departments formed around particular differential. equations mechanics around Lagrange s equation physics around Schro dinger s. equation electromagnetics around Maxwell s equations fluid and gas dynamics. around the Navier Stokes equations solid mechanics around Navier s elasticity. equations nuclear engineering around the transport equation and so on Each. discipline has largely developed its own set of analytic and numerical tools for. attacking its special differential equation independently and this set of tools. forms the basic theoretical core of the discipline and its courses The organi. zation principle reflects both the importance of mathematical modeling using. differential equations and the traditional difficulty of obtaining solutions. Both of these questions would have completely different answers if it were. possible to compute solutions of differential equations using a unified mathe. matical methodology simple enough to be introduced in the basic mathematics. education and powerful enough to apply to real applications In a natural way. Preface xi, mathematics education would then be opened to a wealth of applications and. applied sciences could start from a more practical mathematical foundation. Moreover establishing a common methodology opens the possibility of ex. ploring multi physics problems including the interaction of phenomena from. solids fluids electromagnetics and chemical reactions for example. In this book and the companion volumes we seek to develop such a unified. mathematical methodology for solving differential equations numerically Our. work is based on the conviction that it is possible to approach this area which. is traditionally considered to be difficult and advanced in a way that is com. paratively easy to understand However our goal has not been to write an easy. text that can be covered in one term in an independent course The material in. this book takes time to digest as much as the underlying mathematics itself It. appears to us that the optimal course will involve the gradual integration of the. material into the traditional mathematics curriculum from the very beginning. We emphasize that we are not advocating the study of computational algo. rithms over the mathematics of calculus and linear algebra it is always a fusion. of analysis and numerical computation that appears to be the most fruitful. The material that we would like to see included in the mathematics curriculum. offers a concrete motivation for the development of analytic techniques and. mathematical abstraction Computation does not make analysis obsolete but. gives the analytical mind a focus Furthermore the role of symbolic methods. changes Instead of being the workhorse of analytical computations requiring. a high degree of technical complexity symbolic analysis may focus on analyti. cal aspects of model problems in order to increase understanding and develop. How to use this book, This book begins with a chapter that recalls the close connection between inte. gration and numerical quadrature and then proceeds through introductory ma. terial on calculus and linear algebra to linear ordinary and partial differential. equations The companion text Advanced Computational Differential Equa. tions widens the scope to nonlinear differential equations modeling a variety of. phenomena including reaction diffusion fluid flow and many body dynamics as. well as material on implementation and reaches the frontiers of research The. companion text Introduction to Computational Differential Equations goes in. the other direction developing in detail the introductory material on calculus. and linear algebra, We have used the material that serves as the basis for these books in a. variety of courses in engineering and science taught at the California Insti. tute of Technology Chalmers University of Technology Georgia Institute of. Computational Differential Equations K Eriksson D Estep P Hansbo and C Johnson February 23 2009

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