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

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 1 d 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 1 d 7.1 The Heat Equation in 1d 7.2 The Wave Equation in 1d 7.3 Convection–Diffusion Problems

11 8 Approximation in Several Dimensions8.1 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 ExercisesChapter 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 AssignmentsC.1 Assignment 1 C.2 Assignment 2

17 Appendix D: Appendix DSymbolsSymbolsD.1 Table of Symbols

18 Bibliography

19 Index

20 End User License Agreement

1 Chapter 8Table 8.1 Some one‐dimensional finite elements.Table 8.2 Some two‐dimensional finite elements with triangular elements.Table 8.3 Some two‐dimensional finite elements with quadrilateral elements.Table 8.4 Some three‐dimensional finite elements with tetrahedron elements.

1 Chapter 1 Figure 1.1 Tricomi equation: an example of a variable coefficient classifica... Figure 1.2 Outward unit normal at a point . Figure 1.3 A heat‐conducting one‐dimensional wire. Figure 1.4 A vibrating string.

2 Chapter 2Figure 2.1 The hat function over the interval .Figure 2.2 Illustrating the existence of a unique solution for (V) and (M)....

3 Chapter 3Figure 3.1 Linear Lagrange basis functions for .Figure 3.2 The linear interpolant on a single interval.Figure 3.3 An example of a function in .Figure 3.4 A general piecewise linear basis function .Figure 3.5 A partition of .Figure 3.6 Piecewise linear basis functions.Figure 3.7 and Figure 3.8 and .Figure 3.9 (a) Linear interpolation and (b) basis functions for .Figure 3.10 Linear Lagrange basis functions for .Figure 3.11 Piecewise linear interpolant of .Figure 3.12 Linear Lagrange basis functions for on subinterval .Figure 3.13 Example of a projection onto .Figure 3.14 An example of a function and its