Table of Contents 1
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 13
10 The Initial Boundary Value Problems in 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
