Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 2
Table of Contents
Оглавление1 Cover
2 Preface
4 1 Introduction 1.1 Preliminaries 1.2 Trinities for Second‐Order PDEs 1.3 PDEs in, Further Classifications 1.4 Differential Operators, Superposition 1.5 Some Equations of Mathematical Physics
5 2 Mathematical Tools 2.1 Vector Spaces 2.2 Function Spaces 2.3 Some Basic Inequalities 2.4 Fundamental Solution of PDEs1 2.5 The Weak/Variational Formulation 2.6 A Framework for Analytic Solution in 1d 2.7 An Abstract Framework 2.8 Exercises
6 3 Polynomial Approximation/Interpolation in 1d 3.1 Finite Dimensional Space of Functions on an Interval 3.2 An Ordinary Differential Equation (ODE) 3.3 A Galerkin Method for (BVP) 3.4 Exercises 3.5 Polynomial Interpolation in 1d 3.6 Orthogonal‐ and L2‐Projection 3.7 Numerical Integration, Quadrature Rule 3.8 Exercises
7 4 Linear Systems of Equations 4.1 Direct Methods 4.2 Iterative Methods 4.3 Exercises
8 5 Two‐Point Boundary Value Problems 5.1 The Finite Element Method (FEM) 5.2 Error Estimates in the Energy Norm 5.3 FEM for Convection–Diffusion–Absorption BVPs 5.4 Exercises
9 6 Scalar Initial Value Problems 6.1 Solution Formula and Stability 6.2 Finite Difference Methods for IVP 6.3 Galerkin Finite Element Methods for IVP 6.4 A Posteriori Error Estimates 6.5 A Priori Error Analysis 6.6 The Parabolic Case (a(t) ≥ 0) 6.7 Exercises
10 7 Initial Boundary Value Problems in 1d 7.1 The Heat Equation in 1d 7.2 The Wave Equation in 1d 7.3 Convection–Diffusion Problems
11 8 Approximation in Several Dimensions 8.1 Introduction 8.2 Piecewise Linear Approximation in 2d 8.3 Constructing Finite Element Spaces 8.4 The Interpolant 8.5 The L2 (Revisited) and Ritz Projections 8.6 Exercises
12 9 The Boundary Value Problems in N 9.1 The Poisson Equation 9.2 Stationary Convection–Diffusion Equation 9.3 Hyperbolicity Features 9.4 Exercises
13 10 The Initial Boundary Value Problems in N 10.1 The Heat Equation in N 10.2 The Wave Equation in d 10.3 Exercises
14 Appendix A: Appendix AAnswers to Some ExercisesAnswers to Some Exercises Chapter 1. Exercise Section 1.4.1 Chapter 1. Exercise Section 1.5.4 Chapter 2. Exercise Section 2.11 Chapter 3. Exercise Section 3.5 Chapter 3. Exercise Section 3.8 Chapter 4. Exercise Section 4.3 Chapter 5. Exercise Section 5.4 Chapter 6. Exercise Section 6.7 Chapter 7. Exercise Section 7.2.3 Chapter 7. Exercise Section 7.3.3 Chapter 9. Poisson Equation. Exercise Section 9.4 Chapter 10. IBVPs: Exercise Section 10.3
15 Appendind B: Appendind BAlgorithms and Matlab CodesAlgorithms and Matlab Codes B.1 A Matlab Code to Compute the Mass Matrix M for a Nonuniform Mesh B.2 Matlab Routine to Compute the L2‐Projection B.3 A Matlab Routine Assembling the Stiffness Matrix B.4 A Matlab Routine to Assemble the Convection Matrix B.5 Matlab Routine for Forward‐, Backward‐Euler, and Crank–Nicolson B.6 A Matlab Routine for Mass‐Matrix in 2d B.7 A Matlab Routine for a Poisson Assembler in 2d
16 Appendix C: Appendix CSample AssignmentsSample Assignments C.1 Assignment 1 C.2 Assignment 2
17 Appendix D: Appendix DSymbolsSymbols D.1 Table of Symbols
18 Bibliography
19 Index
