Table of Contents
Preface xi
Chapter 1 Overview 1
Chapter 2 Mathematical Preliminaries 5
2.1 Real Numbers 5
2.2 Functions 5
2.3 Linear Spaces, Linear Mappings, and Bilinear Forms 6
2.4 Linear Independence, Hamel Bases, and Dimension 9
2.5 The Matrix Representation of Linear Mappings and Bilinear Forms 10
2.6 Normed Linear Spaces 12
2.7 Functional and Dual Spaces 13
2.8 Green's Formulas 14
Exercises 15
Comments and References 17
Chapter 3 Finite Element Interpolation 19
3.1 ID Finite Element Interpolation 19
3.1.1 The Global Level 19
3.1.2 The Local Level 24
3.2 Finite Elements 26
3.2.1 Simplicial Lagrange Finite Elements of Type (k) 26
3.2.2 Simplicial Hermite Finite Elements of Type (3) 30
3.2.3 The Raviart-Thomas Finite Element 31
3.2.4 The Nedelec Finite Element 33
3.3 Meshes 34
3.4 Finite Element Spaces and Interpolations 37
3.4.1 H1-Conformal Finite Element Spaces 38
3.4.1.1 Lagrange Elements 39
3.4.1.2 Hermite Elements 43
3.4.2 H(div)-Conformal Finite Element Spaces 43
3.4.3 H(curl)-Conformal Finite Element Spaces 45
3.4.4 Affine Families of Finite Elements 45
3.5 Convergence of Interpolations 46
Exercises 50
Computer Exercises 51
Comments and References 52
Chapter 4 Conforming Finite Element Methods for PDEs 55
4.1 Second-Order Elliptic PDEs 55
4.2 Weak Formulations of Elliptic PDEs 56
4.2.1 Dirichlet Boundary Condition 57
4.2.2 Neumann Boundary Condition 58
4.2.3 Robin Boundary Condition 59
4.3 Well-posedness of Weak Formulations 60
4.4 Variational Structure 61
4.5 The Galerkin Method and Finite Element Methods 62
4.5.1 The Stiffness Matrix 63
4.5.2 Well-posedness of Coercive Discrete Problems 64
4.5.3 Convergence of Finite Element Solutions 64
4.6 Implementation: The Poisson Equation 66
4.6.1 Dirichlet Boundary Condition 66
4.6.2 Mixed Dirichlet-Neumann Boundary Condition 70
4.6.3 Robin Boundary Condition 74
4.7 Time-Dependent Problems: Parabolic Problems 76
4.7.1 Finite Element Approximations using the Method of Lines 78
4.7.2 Temporal Discretization 79
4.7.3 Implementation: A Diffusion Problem 79
4.8 Mixed Finite Element Methods 82
4.8.1 Mixed Formulations 83
4.8.2 Mixed Methods and inf-sup Conditions 84
4.8.3 Implementation 85
Exercises 90
Computer Exercises 92
Comments and References 93
Chapter 5 Applications 97
5.1 Elastic Bars 97
5.2 Euler-Bernoulli Beams 100
5.3 Elastic Membranes 104
5.4 The Wave Equation 105
5.5 Heat Transfer in a Turbine Blade 107
5.6 Seepage in Embankment 112
5.7 Soil Consolidation 117
5.8 The Stokes Equation for Incompressible Fluids 122
5.9 Linearized Elasticity 126
5.10 Linearized Elastodynamics: The Hamburg Wheel-Track Test 129
5.11 Nonlinear Elasticity 134
Exercises 143
Computer Exercises 145
Appendix A FEniCS Installation 147
Appendix B Introduction to Python 149
B.1 Running Python Programs 149
B.2 Lists 150
B.3 Branching and Loops 151
B.4 Functions 152
B.5 Classes and Objects 152
B.6 Reading and Writing Files 154
B.7 Numerical Python Arrays 155
B.8 Plotting with Matplotlib 157
References 159
Index 161
