An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh

      

      Table of Contents

      1  Cover

      2  Preface

      3  Acknowledgments

      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