Master the finite element method with this masterful and practical volume
An Introduction to the Finite Element Method (FEM) for Differential Equations provides readers with a practical and approachable examination of the use of the finite element method in mathematics. Author Mohammad Asadzadeh covers basic FEM theory, both in one-dimensional and higher dimensional cases.
The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, An Introduction to the Finite Element Method covers topics including:
- An introduction to basic ordinary and partial differential equations
- The concept of fundamental solutions using Green's function approaches
- Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations
- Higher-dimensional interpolation procedures
- Stability and convergence analysis of FEM for differential equations
This book is ideal for upper-level undergraduate and graduate students in natural science and engineering. It belongs on the shelf of anyone seeking to improve their understanding of differential equations.
Table of Contents
Preface xi
Acknowledgments xiii
1 Introduction 1
1.1 Preliminaries 2
1.2 Trinities for Second-Order PDEs 4
1.3 PDEs in ℝn, Further Classifications 10
1.4 Differential Operators, Superposition 12
1.4.1 Exercises 14
1.5 Some Equations of Mathematical Physics 15
1.5.1 The Poisson Equation 16
1.5.2 The Heat Equation 17
1.5.2.1 A Model Problem for the Stationary Heat Equation in 1d 17
1.5.2.2 Fourier’s Law of Heat Conduction, Derivation of the Heat Equation 18
1.5.3 The Wave Equation 21
1.5.3.1 The Vibrating String, Derivation of the Wave Equation in 1d 21
1.5.4 Exercises 24
2 Mathematical Tools 27
2.1 Vector Spaces 27
2.1.1 Linear Independence, Basis, and Dimension 30
2.2 Function Spaces 33
2.2.1 Spaces of Differentiable Functions 33
2.2.2 Spaces of Integrable Functions 34
2.2.3 Weak Derivative 35
2.2.4 Sobolev Spaces 36
2.2.5 Hilbert Spaces 37
2.3 Some Basic Inequalities 38
2.4 Fundamental Solution of PDEs 41
2.4.1 Green’s Functions 43
2.5 The Weak/Variational Formulation 44
2.6 A Framework for Analytic Solution in 1d 46
2.6.1 The Variational Formulation in 1d 48
2.6.2 The Minimization Problem in 1d 51
2.6.3 A Mixed Boundary Value Problem in 1d 52
2.7 An Abstract Framework 54
2.7.1 Riesz and Lax-Milgram Theorems 57
2.8 Exercises 63
3 Polynomial Approximation/Interpolation in 1d 67
3.1 Finite Dimensional Space of Functions on an Interval 67
3.2 An Ordinary Differential Equation (ODE) 71
3.2.1 Forward Euler Method to Solve IVP 71
3.2.2 Variational Formulation for IVP 72
3.2.3 Galerkin Method for IVP 73
3.3 A Galerkin Method for (BVP) 74
3.3.1 An Equivalent Finite Difference Approach 79
3.4 Exercises 82
3.5 Polynomial Interpolation in 1d 83
3.5.1 Lagrange Interpolation 90
3.6 Orthogonal- and L2-Projection 94
3.6.1 The L2-Projection onto the Space of Polynomials 94
3.7 Numerical Integration, Quadrature Rule 96
3.7.1 Composite Rules for Uniform Partitions 98
3.7.2 Gauss Quadrature Rule 101
3.8 Exercises 105
4 Linear Systems of Equations 109
4.1 Direct Methods 110
4.1.1 LU Factorization of an n × n Matrix A 113
4.2 Iterative Methods 115
4.2.1 Jacobi Iteration 115
4.2.2 Convergence Criterion 116
4.2.3 Gauss-Seidel Iteration 117
4.2.4 The Successive Over-Relaxation Method (S.O.R.) 119
4.2.5 Abstraction of Iterative Methods 120
4.2.5.1 Questions 120
4.2.6 Jacobi’s Method 120
4.2.7 Gauss-Seidel’s Method 121
4.2.7.1 Relaxation 121
4.3 Exercises 122
5 Two-Point Boundary Value Problems 125
5.1 The Finite Element Method (FEM) 125
5.2 Error Estimates in the Energy Norm 127
5.2.1 Adaptivity 132
5.3 FEM for Convection-Diffusion-Absorption BVPs 132
5.4 Exercises 140
6 Scalar Initial Value Problems 147
6.1 Solution Formula and Stability 147
6.2 Finite Difference Methods for IVP 149
6.3 Galerkin Finite Element Methods for IVP 151
6.3.1 The Continuous Galerkin Method 152
6.3.1.1 The cG(1) Algorithm 154
6.3.1.2 The cG(q) Method 154
6.3.2 The Discontinuous Galerkin Method 155
6.4 A Posteriori Error Estimates 156
6.4.1 A Posteriori Error Estimate for cG(1) 156
6.4.1.1 The Dual Problem 157
6.4.2 A Posteriori Error Estimate for dG(0) 161
6.4.3 Adaptivity for dG(0) 163
6.4.3.1 An Adaptivity Algorithm 163
6.5 A Priori Error Analysis 164
6.5.1 A Priori Error Estimates for the dG(0) Method 164
6.6 The Parabolic Case (a(t) ≥ 0) 168
6.6.1 An Example of Error Estimate 171
6.7 Exercises 173
7 Initial Boundary Value Problems in 1d 177
7.1 The Heat Equation in 1d 177
7.1.1 Stability Estimates 179
7.1.2 FEM for the Heat Equation 183
7.1.3 Error Analysis 186
7.1.4 Exercises 192
7.2 The Wave Equation in 1d 193
7.2.1 Wave Equation as a System of Hyperbolic PDEs 194
7.2.2 The Finite Element Discretization Procedure 195
7.2.3 Exercises 197
7.3 Convection-Diffusion Problems 199
7.3.1 Finite Element Method 202
7.3.2 The Streamline-Diffusion Method (SDM) 203
7.3.3 Exercises 205
8 Approximation in Several Dimensions 207
8.1 Introduction 207
8.2 Piecewise Linear Approximation in 2d 209
8.2.1 Basis Functions for the Piecewise Linears in 2d 209
8.3 Constructing Finite Element Spaces 216
8.4 The Interpolant 219
8.4.1 Error Estimates for Piecewise Linear Interpolation 222
8.5 The L2 (Revisited) and Ritz Projections 228
8.5.1 The Ritz or Elliptic Projection 230
8.6 Exercises 231
9 The Boundary Value Problems in ℝN 235
9.1 The Poisson Equation 235
9.1.1 Weak Stability 236
9.1.2 Error Estimates for the CG(1) FEM 237
9.1.3 Proof of the Regularity Lemma 242
9.2 Stationary Convection-Diffusion Equation 243
9.2.1 The Elliptic Case 243
9.2.1.1 A Brief Note on Distributions 244
9.2.2 Error Estimates 248
9.3 Hyperbolicity Features 249
9.3.1 Convection Dominating Case 250
9.3.2 The SD Method for Convection Diffusion Problem 251
9.3.3 Stability Estimates 252
9.3.4 Error Estimates for Convention Dominating in 2d 252
9.4 Exercises 255
10 The Initial Boundary Value Problems in ℝN 261
10.1 The Heat Equation in ℝN261
10.1.1 The Fundamental Solution 262
10.1.2 Stability 263
10.1.3 The Finite Element for Heat Equation 265
10.1.3.1 The Semidiscrete Problem 265
10.1.4 A Fully Discrete Algorithm 269
10.1.5 The Discrete Equations 270
10.1.6 A Priori Error Estimate: Fully Discrete Problem 271
10.2 The Wave Equation in ℝd 272
10.2.1 The Weak Formulation 273
10.2.2 The Semidiscrete Problem 273
10.2.2.1 A Priori Error Estimates for the Semidiscrete Problem 274
10.2.3 The Fully Discrete Problem 275
10.2.3.1 Finite Elements for the Fully Discrete Problem 276
10.2.4 Error Estimate for cG(1) 278
10.3 Exercises 279
Appendix A Answers to Some Exercises 285
Chapter 1. Exercise Section 1.4.1 285
Chapter 1. Exercise Section 1.5.4 285
Chapter 2. Exercise Section 2.11 286
Chapter 3. Exercise Section 3.5 286
Chapter 3. Exercise Section 3.8 287
Chapter 4. Exercise Section 4.3 288
Chapter 5. Exercise Section 5.4 289
Chapter 6. Exercise Section 6.7 291
Chapter 7. Exercise Section 7.2.3 292
Chapter 7. Exercise Section 7.3.3 292
Chapter 9. Poisson Equation. Exercise Section 9.4 292
Chapter 10. IBVPs: Exercise Section 10.3 293
Appendix B Algorithms and Matlab Codes 295
B.1 A Matlab Code to Compute the Mass Matrix M for a Nonuniform Mesh 296
B.1.1 A Matlab Routine to Compute the Load Vector b 297
B.2 Matlab Routine to Compute the L2-Projection 298
B.2.1 A Matlab Routine for the Composite Midpoint Rule 299
B.2.2 A Matlab Routine for the Composite Trapezoidal Rule 299
B.2.3 A Matlab Routine for the Composite Simpson’s Rule 299
B.3 A Matlab Routine Assembling the Stiffness Matrix 300
B.4 A Matlab Routine to Assemble the Convection Matrix 301
B.5 Matlab Routine for Forward-, Backward-Euler, and Crank-Nicolson 302
B.6 A Matlab Routine for Mass-Matrix in 2d 304
B.7 A Matlab Routine for a Poisson Assembler in 2d 304
Appendix C Sample Assignments 307
C.1 Assignment 1 307
C.2 Assignment 2 308
C.2.1 Grading Policy of the Assignment 308
C.2.2 Theory 308
C.2.3 Selected Applications 309
C.2.3.1 Convection-Diffusion-Absorption/Reaction 309
C.2.3.2 Electrostatics 310
C.2.3.3 2d Fluid Flow 310
C.2.3.4 Heat Conduction 310
C.2.3.5 Quantum Physics 310
Appendix D Symbols 313
D.1 Table of Symbols 313
Bibliography 317
Index 327
