Numerical Methods of Scientific Computing [2 ed.] 9065623639
302 28 127MB
English Pages 154 Year 2014
Table of contents :
Contents
Chapter 1 Review of some basic mathematical concepts
1.1 Preliminaries
1.2 Global contents of the book
1.3 Building blocks for mathematical modeling
1.3.1 Gradient of a scalar
1.3.2 Directional derivative
1.3.3 Divergence of a vector field
1.3.4 Gauss' divergence theorem
1.3.5 Conservation laws
1.4 Minimization
1.4.1 Elastic string
1.5 Preliminaries from linear algebra
1.6 Some theorems used in the mathematical theory
1.7 Summary of Chapter 1
Chapter 2 A crash course in PDE's
Chapter 3 Finite difference methods
Chapter 4 Finite volume methods
Chapter 5 Minimization problems in physics
5.1 Introduction
5.1.1 Minimal potential energy
5.1.2 Derivation of the differential equation
5.2 A general 1-D problem with 1st order derivatives
5.3 A simple 2-D case
5.4 Examples of minimization problems
5.4.1 Minimal surface problem
5.4.2 Minimal potential energy
5.4.3 Small displacement theory of elasticity (Plane stress)
5.4.4 Loaded and clamped plate
5.5 A 2-D problem
5.6 Theorectical remarks
5.6.1 Smoothness requirements
5.6.2 Boundary conditions
5.6.3 Weak formulation
5.7 Exercises
5.8 From PDE to minimization problem
5.8.1 Introduction
5.8.2 Linear problems with homogeneous boundary conditions
5.8.3 Linear problems with non-homogeneous boundary conditions
5.8.4 Exercises
5.9 Mathematical theory of minimization
5.10 Summary of chapter 5
Chapter 6 The numerical solution of minimization problems
6.1 Ritz's method
6.2 The finite element method in R1
6.3 The finite element method in R2
6.4 Theoretical remarks
6.5 Summary of Chapter 6
Chapter 7 The weak formulation and Galerkin's method
7.1 The weak formulation for a symmetrical problem
7.2 The weak formulation for a non-symmetric problem
7.3 Galerlin's method
7.4 Petrov-Galerkin
7.5 An example of a system of coupled PDEs
7.6 Mathematical theory
7.7 Summary of chapter 7
Chapter 8 Extension of the FEM
Chapter 9 Solution of large systems of equations
9.7 Non-linear equations
Chapter 10 The heat- or diffusion equation
Chapter 11 The wave equation
Chapter 12 The transport equation
Chapter 13 Moving boundary problems
Bibliography
Contents
Chapter 1 Review of some basic mathematical concepts
1.1 Preliminaries
1.2 Global contents of the book
1.3 Building blocks for mathematical modeling
1.3.1 Gradient of a scalar
1.3.2 Directional derivative
1.3.3 Divergence of a vector field
1.3.4 Gauss' divergence theorem
1.3.5 Conservation laws
1.4 Minimization
1.4.1 Elastic string
1.5 Preliminaries from linear algebra
1.6 Some theorems used in the mathematical theory
1.7 Summary of Chapter 1
Chapter 2 A crash course in PDE's
Chapter 3 Finite difference methods
Chapter 4 Finite volume methods
Chapter 5 Minimization problems in physics
5.1 Introduction
5.1.1 Minimal potential energy
5.1.2 Derivation of the differential equation
5.2 A general 1-D problem with 1st order derivatives
5.3 A simple 2-D case
5.4 Examples of minimization problems
5.4.1 Minimal surface problem
5.4.2 Minimal potential energy
5.4.3 Small displacement theory of elasticity (Plane stress)
5.4.4 Loaded and clamped plate
5.5 A 2-D problem
5.6 Theorectical remarks
5.6.1 Smoothness requirements
5.6.2 Boundary conditions
5.6.3 Weak formulation
5.7 Exercises
5.8 From PDE to minimization problem
5.8.1 Introduction
5.8.2 Linear problems with homogeneous boundary conditions
5.8.3 Linear problems with non-homogeneous boundary conditions
5.8.4 Exercises
5.9 Mathematical theory of minimization
5.10 Summary of chapter 5
Chapter 6 The numerical solution of minimization problems
6.1 Ritz's method
6.2 The finite element method in R1
6.3 The finite element method in R2
6.4 Theoretical remarks
6.5 Summary of Chapter 6
Chapter 7 The weak formulation and Galerkin's method
7.1 The weak formulation for a symmetrical problem
7.2 The weak formulation for a non-symmetric problem
7.3 Galerlin's method
7.4 Petrov-Galerkin
7.5 An example of a system of coupled PDEs
7.6 Mathematical theory
7.7 Summary of chapter 7
Chapter 8 Extension of the FEM
Chapter 9 Solution of large systems of equations
9.7 Non-linear equations
Chapter 10 The heat- or diffusion equation
Chapter 11 The wave equation
Chapter 12 The transport equation
Chapter 13 Moving boundary problems
Bibliography
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