Preface page ................................................. xv
1 Introduction to Partial Differential Equations .............. 1
1.1 Types of Second-Order PDEs ............................. 1
1.2 Physical Problems ...................................... 2
1.3 Summary ............................................... 10
2 Parabolic Equations ........................................ 13
2.1 Theory of Linear Parabolic Equations .................. 13
2.2 Finite Difference Methods in One Dimension ............ 21
2.3 Lax Convergence Theorem ............................... 59
2.4 Fourier Analysis ...................................... 60
2.3 Diffusion Problems .................................... 63
2.5 Lax Equivalence Theorem ............................... 70
2.6 Measuring Accuracy and Efficiency ..................... 74
2.7 Finite Difference Methods in Multiple Dimensions ...... 78
3 Iterative Linear Algebra ................................... 85
3.1 Relative Efficiency of Implicit Computations .......... 85
3.2 Vector Norms .......................................... 89
3.3 Matrix Norms .......................................... 90
3.4 Neumann Series ........................................ 94
3.5 Perron-Frobenius Theorem .............................. 96
3.6 M-Matrices ............................................ 98
3.7 Iterative Improvement ................................ 103
3.8 Gradient Methods ..................................... 123
3.9 Minimum Residual Methods ............................. 142
3.10 Nonlinear Systems .................................... 153
3.11 Multigrid ............................................ 158
4 Introduction to Finite Element Methods .................... 179
4.1 Weak Formulation ..................................... 179
4.2 Applications ......................................... 183
4.3 Galerkin Methods ..................................... 188
4.4 Finite Element Example ............................... 190
4.5 Overview of Finite Elements .......................... 200
4.6 Reference Shapes ..................................... 202
4.7 Polynomial Families .................................. 210
4.8 Multi-Indices ........................................ 217
4.9 Shape Function Families .............................. 219
4.10 Quadrature Rules ..................................... 225
4.11 Mesh Generation ...................................... 237
4.12 Coordinate Mappings .................................. 238
4.13 Finite Elements ...................................... 253
4.14 Linear Systems ....................................... 253
5 Finite Element Theory ..................................... 263
5.1 Norms and Derivatives ................................ 263
5.2 Sobolev Spaces ....................................... 274
5.3 Elliptic Equations ................................... 289
5.4 Elliptic Regularity .................................. 300
5.5 Galerkin Methods ..................................... 313
6 Finite Element Approximations ............................. 332
6.1 Gaps in Our Theory ................................... 332
6.2 Finite Element Assumptions ........................... 333
6.3 Piecewise Polynomial Approximation ................... 336
6.4 Conforming Spaces .................................... 342
6.5 Useful Approximations ................................ 366
6.6 Refinement ........................................... 386
6.7 Inverse Estimates .................................... 388
6.8 Condition Number Estimates ........................... 389
7 Mixed and Hybrid Finite Elements .......................... 398
7.1 Hdiv and Hcurl ....................................... 399
7.2 Physical Problems .................................... 401
7.3 Saddle-Point Problems ................................ 416
7.4 Mixed Finite Elements ................................ 431
7.5 Iterative Methods .................................... 505
7.6 Hybrid Mixed Finite Elements ......................... 514
8 Finite Elements for Parabolic Equations ................... 520
8.1 Well-Posedness ....................................... 520
8.2 Galerkin Methods ..................................... 528
8.3 Convection-Diffusion Problems ........................ 541
8.4 Reaction-Diffusion Problems .......................... 552
9 Finite Elements and Multigrid ............................. 554
9.1 Assumptions .......................................... 554
9.2 Prolongation and Restriction ......................... 555
9.3 Coarse Grid Projection ............................... 562
9.4 Parabolic Problems ................................... 562
9.5 Mixed Methods ........................................ 563
10 Local Refinement .......................................... 564
10.1 Locally Refined Tessellations ........................ 564
10.2 Clement's Interpolation .............................. 567
10.3 Bubble Functions ..................................... 570
10.4 Residual Estimator ................................... 574
10.5 Other Error Estimators ............................... 579
10.6 Adaptive Mesh Refinement ............................. 580
10.7 Mortar Methods ....................................... 581
Nomenclature ................................................. 610
References ................................................... 616
Author index ................................................. 628
Subject index ................................................ 631
| 
