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Contents
Series Preface v
Preface vii
PART I: Getting Started
1. Foundations of Matrix Analysis 1
1.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Operations with Matrices . . . . . . . . . . . . . . . . . . . 5
1.3.1 Inverse of a Matrix . . . . . . . . . . . . . . . . . . 6
1.3.2 Matrices and Linear Mappings . . . . . . . . . . . 7
1.3.3 Operations with Block-Partitioned Matrices . . . . 7
1.4 Trace and Determinant of a Matrix . . . . . . . . . . . . . 8
1.5 Rank and Kernel of a Matrix . . . . . . . . . . . . . . . . 9
1.6 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.1 Block Diagonal Matrices . . . . . . . . . . . . . . . 10
1.6.2 Trapezoidal and Triangular Matrices . . . . . . . . 11
1.6.3 Banded Matrices . . . . . . . . . . . . . . . . . . . 11
1.7 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 12
1.8 Similarity Transformations . . . . . . . . . . . . . . . . . . 14
1.9 The Singular Value Decomposition (SVD) . . . . . . . . . 16
1.10 Scalar Product and Norms in Vector Spaces . . . . . . . . 17
1.11 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . 21xii Contents
1.11.1 Relation Between Norms and the
Spectral Radius of a Matrix . . . . . . . . . . . . . 25
1.11.2 Sequences and Series of Matrices . . . . . . . . . . 26
1.12 Positive Definite, Diagonally Dominant and M-Matrices . 27
1.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2. Principles of Numerical Mathematics 33
2.1 Well-Posedness and Condition Number of a Problem . . . 33
2.2 Stability of Numerical Methods . . . . . . . . . . . . . . . 37
2.2.1 Relations Between Stability and Convergence . . . 40
2.3 A priori and a posteriori Analysis . . . . . . . . . . . . . . 41
2.4 Sources of Error in Computational Models . . . . . . . . . 43
2.5 Machine Representation of Numbers . . . . . . . . . . . . 45
2.5.1 The Positional System . . . . . . . . . . . . . . . . 45
2.5.2 The Floating-Point Number System . . . . . . . . 46
2.5.3 Distribution of Floating-Point Numbers . . . . . . 49
2.5.4 IEC/IEEE Arithmetic . . . . . . . . . . . . . . . . 49
2.5.5 Rounding of a Real Number in Its
Machine Representation . . . . . . . . . . . . . . . 50
2.5.6 Machine Floating-Point Operations . . . . . . . . . 52
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
PART II: Numerical Linear Algebra
3. Direct Methods for the Solution of Linear Systems 57
3.1 Stability Analysis of Linear Systems . . . . . . . . . . . . 58
3.1.1 The Condition Number of a Matrix . . . . . . . . 58
3.1.2 Forward a priori Analysis . . . . . . . . . . . . . . 60
3.1.3 Backward a priori Analysis . . . . . . . . . . . . . 63
3.1.4 A posteriori Analysis . . . . . . . . . . . . . . . . . 64
3.2 Solution of Triangular Systems . . . . . . . . . . . . . . . 65
3.2.1 Implementation of Substitution Methods . . . . . 65
3.2.2 Rounding Error Analysis . . . . . . . . . . . . . . 67
3.2.3 Inverse of a Triangular Matrix . . . . . . . . . . . 67
3.3 The Gaussian Elimination Method (GEM) and
LU Factorization . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 GEM as a Factorization Method . . . . . . . . . . 72
3.3.2 The Effect of Rounding Errors . . . . . . . . . . . 76
3.3.3 Implementation of LU Factorization . . . . . . . . 77
3.3.4 Compact Forms of Factorization . . . . . . . . . . 78
3.4 Other Types of Factorization . . . . . . . . . . . . . . . . . 79
3.4.1 LDMT
Factorization . . . . . . . . . . . . . . . . . 79
3.4.2 Symmetric and Positive Definite Matrices:
The Cholesky Factorization . . . . . . . . . . . . . 80
3.4.3 Rectangular Matrices: The QR Factorization . . . 82Contents xiii
3.5 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.6 Computing the Inverse of a Matrix . . . . . . . . . . . . . 89
3.7 Banded Systems . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7.1 Tridiagonal Matrices . . . . . . . . . . . . . . . . . 91
3.7.2 Implementation Issues . . . . . . . . . . . . . . . . 92
3.8 Block Systems . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.8.1 Block LU Factorization . . . . . . . . . . . . . . . 94
3.8.2 Inverse of a Block-Partitioned Matrix . . . . . . . 95
3.8.3 Block Tridiagonal Systems . . . . . . . . . . . . . . 95
3.9 Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . 97
3.9.1 The Cuthill-McKee Algorithm . . . . . . . . . . . 98
3.9.2 Decomposition into Substructures . . . . . . . . . 100
3.9.3 Nested Dissection . . . . . . . . . . . . . . . . . . . 103
3.10 Accuracy of the Solution Achieved Using GEM . . . . . . 103
3.11 An Approximate Computation of K(A) . . . . . . . . . . . 106
3.12 Improving the Accuracy of GEM . . . . . . . . . . . . . . 109
3.12.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . 110
3.12.2 Iterative Refinement . . . . . . . . . . . . . . . . . 111
3.13 Undetermined Systems . . . . . . . . . . . . . . . . . . . . 112
3.14 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.14.1 Nodal Analysis of a Structured Frame . . . . . . . 115
3.14.2 Regularization of a Triangular Grid . . . . . . . . 118
3.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4. Iterative Methods for Solving Linear Systems 123
4.1 On the Convergence of Iterative Methods . . . . . . . . . . 123
4.2 Linear Iterative Methods . . . . . . . . . . . . . . . . . . . 126
4.2.1 Jacobi, Gauss-Seidel and Relaxation Methods . . . 127
4.2.2 Convergence Results for Jacobi and
Gauss-Seidel Methods . . . . . . . . . . . . . . . . 129
4.2.3 Convergence Results for the Relaxation Method . 131
4.2.4 A priori Forward Analysis . . . . . . . . . . . . . . 132
4.2.5 Block Matrices . . . . . . . . . . . . . . . . . . . . 133
4.2.6 Symmetric Form of the Gauss-Seidel and
SOR Methods . . . . . . . . . . . . . . . . . . . . . 133
4.2.7 Implementation Issues . . . . . . . . . . . . . . . . 135
4.3 Stationary and Nonstationary Iterative Methods . . . . . . 136
4.3.1 Convergence Analysis of the Richardson Method . 137
4.3.2 Preconditioning Matrices . . . . . . . . . . . . . . 139
4.3.3 The Gradient Method . . . . . . . . . . . . . . . . 146
4.3.4 The Conjugate Gradient Method . . . . . . . . . . 150
4.3.5 The Preconditioned Conjugate Gradient Method . 156
4.3.6 The Alternating-Direction Method . . . . . . . . . 158
4.4 Methods Based on Krylov Subspace Iterations . . . . . . . 159
4.4.1 The Arnoldi Method for Linear Systems . . . . . . 162xiv Contents
4.4.2 The GMRES Method . . . . . . . . . . . . . . . . 165
4.4.3 The Lanczos Method for Symmetric Systems . . . 167
4.5 The Lanczos Method for Unsymmetric Systems . . . . . . 168
4.6 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . 171
4.6.1 A Stopping Test Based on the Increment . . . . . 172
4.6.2 A Stopping Test Based on the Residual . . . . . . 174
4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.7.1 Analysis of an Electric Network . . . . . . . . . . . 174
4.7.2 Finite Difference Analysis of Beam Bending . . . . 177
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5. Approximation of Eigenvalues and Eigenvectors 183
5.1 Geometrical Location of the Eigenvalues . . . . . . . . . . 183
5.2 Stability and Conditioning Analysis . . . . . . . . . . . . . 186
5.2.1 A priori Estimates . . . . . . . . . . . . . . . . . . 186
5.2.2 A posteriori Estimates . . . . . . . . . . . . . . . . 190
5.3 The Power Method . . . . . . . . . . . . . . . . . . . . . . 192
5.3.1 Approximation of the Eigenvalue of
Largest Module . . . . . . . . . . . . . . . . . . . . 192
5.3.2 Inverse Iteration . . . . . . . . . . . . . . . . . . . 195
5.3.3 Implementation Issues . . . . . . . . . . . . . . . . 196
5.4 The QR Iteration . . . . . . . . . . . . . . . . . . . . . . . 200
5.5 The Basic QR Iteration . . . . . . . . . . . . . . . . . . . . 201
5.6 The QR Method for Matrices in Hessenberg Form . . . . . 203
5.6.1 Householder and Givens Transformation Matrices 204
5.6.2 Reducing a Matrix in Hessenberg Form . . . . . . 207
5.6.3 QR Factorization of a Matrix in Hessenberg Form 209
5.6.4 The Basic QR Iteration Starting from
Upper Hessenberg Form . . . . . . . . . . . . . . . 210
5.6.5 Implementation of Transformation Matrices . . . . 212
5.7 The QR Iteration with Shifting Techniques . . . . . . . . . 215
5.7.1 The QR Method with Single Shift . . . . . . . . . 215
5.7.2 The QR Method with Double Shift . . . . . . . . . 218
5.8 Computing the Eigenvectors and the SVD of a Matrix . . 221
5.8.1 The Hessenberg Inverse Iteration . . . . . . . . . . 221
5.8.2 Computing the Eigenvectors from the
Schur Form of a Matrix . . . . . . . . . . . . . . . 221
5.8.3 Approximate Computation of the SVD of a Matrix 222
5.9 The Generalized Eigenvalue Problem . . . . . . . . . . . . 224
5.9.1 Computing the Generalized Real Schur Form . . . 225
5.9.2 Generalized Real Schur Form of
Symmetric-Definite Pencils . . . . . . . . . . . . . 226
5.10 Methods for Eigenvalues of Symmetric Matrices . . . . . . 227
5.10.1 The Jacobi Method . . . . . . . . . . . . . . . . . 227
5.10.2 The Method of Sturm Sequences . . . . . . . . . . 230Contents xv
5.11 The Lanczos Method . . . . . . . . . . . . . . . . . . . . . 233
5.12 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.12.1 Analysis of the Buckling of a Beam . . . . . . . . . 236
5.12.2 Free Dynamic Vibration of a Bridge . . . . . . . . 238
5.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
PART III: Around Functions and Functionals
6. Rootfinding for Nonlinear Equations 245
6.1 Conditioning of a Nonlinear Equation . . . . . . . . . . . . 246
6.2 A Geometric Approach to Rootfinding . . . . . . . . . . . 248
6.2.1 The Bisection Method . . . . . . . . . . . . . . . . 248
6.2.2 The Methods of Chord, Secant and Regula Falsi
and Newton’s Method . . . . . . . . . . . . . . . . 251
6.2.3 The Dekker-Brent Method . . . . . . . . . . . . . 256
6.3 Fixed-Point Iterations for Nonlinear Equations . . . . . . . 257
6.3.1 Convergence Results for
Some Fixed-Point Methods . . . . . . . . . . . . . 260
6.4 Zeros of Algebraic Equations . . . . . . . . . . . . . . . . . 261
6.4.1 The Horner Method and Deflation . . . . . . . . . 262
6.4.2 The Newton-Horner Method . . . . . . . . . . . . 263
6.4.3 The Muller Method . . . . . . . . . . . . . . . . . 267
6.5 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . 269
6.6 Post-Processing Techniques for Iterative Methods . . . . . 272
6.6.1 Aitken’s Acceleration . . . . . . . . . . . . . . . . 272
6.6.2 Techniques for Multiple Roots . . . . . . . . . . . 275
6.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.7.1 Analysis of the State Equation for a Real Gas . . 276
6.7.2 Analysis of a Nonlinear Electrical Circuit . . . . . 277
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7. Nonlinear Systems and Numerical Optimization 281
7.1 Solution of Systems of Nonlinear Equations . . . . . . . . 282
7.1.1 Newton’s Method and Its Variants . . . . . . . . . 283
7.1.2 Modified Newton’s Methods . . . . . . . . . . . . . 284
7.1.3 Quasi-Newton Methods . . . . . . . . . . . . . . . 288
7.1.4 Secant-Like Methods . . . . . . . . . . . . . . . . . 288
7.1.5 Fixed-Point Methods . . . . . . . . . . . . . . . . . 290
7.2 Unconstrained Optimization . . . . . . . . . . . . . . . . . 294
7.2.1 Direct Search Methods . . . . . . . . . . . . . . . . 295
7.2.2 Descent Methods . . . . . . . . . . . . . . . . . . . 300
7.2.3 Line Search Techniques . . . . . . . . . . . . . . . 302
7.2.4 Descent Methods for Quadratic Functions . . . . . 304
7.2.5 Newton-Like Methods for Function Minimization . 307
7.2.6 Quasi-Newton Methods . . . . . . . . . . . . . . . 308xvi Contents
7.2.7 Secant-Like Methods . . . . . . . . . . . . . . . . . 309
7.3 Constrained Optimization . . . . . . . . . . . . . . . . . . 311
7.3.1 Kuhn-Tucker Necessary Conditions for
Nonlinear Programming . . . . . . . . . . . . . . . 313
7.3.2 The Penalty Method . . . . . . . . . . . . . . . . . 315
7.3.3 The Method of Lagrange Multipliers . . . . . . . . 317
7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.4.1 Solution of a Nonlinear System Arising from
Semiconductor Device Simulation . . . . . . . . . . 320
7.4.2 Nonlinear Regularization of a Discretization Grid . 323
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
8. Polynomial Interpolation 327
8.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . 328
8.1.1 The Interpolation Error . . . . . . . . . . . . . . . 329
8.1.2 Drawbacks of Polynomial Interpolation on Equally
Spaced Nodes and Runge’s Counterexample . . . . 330
8.1.3 Stability of Polynomial Interpolation . . . . . . . . 332
8.2 Newton Form of the Interpolating Polynomial . . . . . . . 333
8.2.1 Some Properties of Newton Divided Differences . . 335
8.2.2 The Interpolation Error Using Divided Differences 337
8.3 Piecewise Lagrange Interpolation . . . . . . . . . . . . . . 338
8.4 Hermite-Birkoff Interpolation . . . . . . . . . . . . . . . . 341
8.5 Extension to the Two-Dimensional Case . . . . . . . . . . 343
8.5.1 Polynomial Interpolation . . . . . . . . . . . . . . 343
8.5.2 Piecewise Polynomial Interpolation . . . . . . . . . 344
8.6 Approximation by Splines . . . . . . . . . . . . . . . . . . 348
8.6.1 Interpolatory Cubic Splines . . . . . . . . . . . . . 349
8.6.2 B-Splines . . . . . . . . . . . . . . . . . . . . . . . 353
8.7 Splines in Parametric Form . . . . . . . . . . . . . . . . . 357
8.7.1 B´ezier Curves and Parametric B-Splines . . . . . . 359
8.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 362
8.8.1 Finite Element Analysis of a Clamped Beam . . . 363
8.8.2 Geometric Reconstruction Based on
Computer Tomographies . . . . . . . . . . . . . . . 366
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
9. Numerical Integration 371
9.1 Quadrature Formulae . . . . . . . . . . . . . . . . . . . . . 371
9.2 Interpolatory Quadratures . . . . . . . . . . . . . . . . . . 373
9.2.1 The Midpoint or Rectangle Formula . . . . . . . . 373
9.2.2 The Trapezoidal Formula . . . . . . . . . . . . . . 375
9.2.3 The Cavalieri-Simpson Formula . . . . . . . . . . . 377
9.3 Newton-Cotes Formulae . . . . . . . . . . . . . . . . . . . 378
9.4 Composite Newton-Cotes Formulae . . . . . . . . . . . . . 383Contents xvii
9.5 Hermite Quadrature Formulae . . . . . . . . . . . . . . . . 386
9.6 Richardson Extrapolation . . . . . . . . . . . . . . . . . . 387
9.6.1 Romberg Integration . . . . . . . . . . . . . . . . . 389
9.7 Automatic Integration . . . . . . . . . . . . . . . . . . . . 391
9.7.1 Non Adaptive Integration Algorithms . . . . . . . 392
9.7.2 Adaptive Integration Algorithms . . . . . . . . . . 394
9.8 Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . 398
9.8.1 Integrals of Functions with Finite
Jump Discontinuities . . . . . . . . . . . . . . . . . 398
9.8.2 Integrals of Infinite Functions . . . . . . . . . . . . 398
9.8.3 Integrals over Unbounded Intervals . . . . . . . . . 401
9.9 Multidimensional Numerical Integration . . . . . . . . . . 402
9.9.1 The Method of Reduction Formula . . . . . . . . . 403
9.9.2 Two-Dimensional Composite Quadratures . . . . . 404
9.9.3 Monte Carlo Methods for
Numerical Integration . . . . . . . . . . . . . . . . 407
9.10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 408
9.10.1 Computation of an Ellipsoid Surface . . . . . . . . 408
9.10.2 Computation of the Wind Action on a
Sailboat Mast . . . . . . . . . . . . . . . . . . . . . 410
9.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
PART IV: Transforms, Differentiation
and Problem Discretization
10. Orthogonal Polynomials in Approximation Theory 415
10.1 Approximation of Functions by Generalized Fourier Series 415
10.1.1 The Chebyshev Polynomials . . . . . . . . . . . . . 417
10.1.2 The Legendre Polynomials . . . . . . . . . . . . . 419
10.2 Gaussian Integration and Interpolation . . . . . . . . . . . 419
10.3 Chebyshev Integration and Interpolation . . . . . . . . . . 424
10.4 Legendre Integration and Interpolation . . . . . . . . . . . 426
10.5 Gaussian Integration over Unbounded Intervals . . . . . . 428
10.6 Programs for the Implementation of Gaussian Quadratures 429
10.7 Approximation of a Function in the Least-Squares Sense . 431
10.7.1 Discrete Least-Squares Approximation . . . . . . . 431
10.8 The Polynomial of Best Approximation . . . . . . . . . . . 433
10.9 Fourier Trigonometric Polynomials . . . . . . . . . . . . . 435
10.9.1 The Gibbs Phenomenon . . . . . . . . . . . . . . . 439
10.9.2 The Fast Fourier Transform . . . . . . . . . . . . . 440
10.10 Approximation of Function Derivatives . . . . . . . . . . . 442
10.10.1 Classical Finite Difference Methods . . . . . . . . . 442
10.10.2 Compact Finite Differences . . . . . . . . . . . . . 444
10.10.3 Pseudo-Spectral Derivative . . . . . . . . . . . . . 448
10.11 Transforms and Their Applications . . . . . . . . . . . . . 450xviii Contents
10.11.1 The Fourier Transform . . . . . . . . . . . . . . . . 450
10.11.2 (Physical) Linear Systems and Fourier Transform . 453
10.11.3 The Laplace Transform . . . . . . . . . . . . . . . 455
10.11.4 The Z-Transform . . . . . . . . . . . . . . . . . . . 457
10.12 The Wavelet Transform . . . . . . . . . . . . . . . . . . . . 458
10.12.1 The Continuous Wavelet Transform . . . . . . . . 458
10.12.2 Discrete and Orthonormal Wavelets . . . . . . . . 461
10.13 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 463
10.13.1 Numerical Computation of Blackbody Radiation . 463
10.13.2 Numerical Solution of Schr¨odinger Equation . . . . 464
10.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
11. Numerical Solution of Ordinary Differential Equations 469
11.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . 469
11.2 One-Step Numerical Methods . . . . . . . . . . . . . . . . 472
11.3 Analysis of One-Step Methods . . . . . . . . . . . . . . . . 473
11.3.1 The Zero-Stability . . . . . . . . . . . . . . . . . . 475
11.3.2 Convergence Analysis . . . . . . . . . . . . . . . . 477
11.3.3 The Absolute Stability . . . . . . . . . . . . . . . . 479
11.4 Difference Equations . . . . . . . . . . . . . . . . . . . . . 482
11.5 Multistep Methods . . . . . . . . . . . . . . . . . . . . . . 487
11.5.1 Adams Methods . . . . . . . . . . . . . . . . . . . 490
11.5.2 BDF Methods . . . . . . . . . . . . . . . . . . . . 492
11.6 Analysis of Multistep Methods . . . . . . . . . . . . . . . . 492
11.6.1 Consistency . . . . . . . . . . . . . . . . . . . . . . 493
11.6.2 The Root Conditions . . . . . . . . . . . . . . . . . 494
11.6.3 Stability and Convergence Analysis for
Multistep Methods . . . . . . . . . . . . . . . . . . 495
11.6.4 Absolute Stability of Multistep Methods . . . . . . 499
11.7 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . 502
11.8 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . 508
11.8.1 Derivation of an Explicit RK Method . . . . . . . 511
11.8.2 Stepsize Adaptivity for RK Methods . . . . . . . . 512
11.8.3 Implicit RK Methods . . . . . . . . . . . . . . . . 514
11.8.4 Regions of Absolute Stability for RK Methods . . 516
11.9 Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . 517
11.10 Stiff Problems . . . . . . . . . . . . . . . . . . . . . . . . . 519
11.11 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 521
11.11.1 Analysis of the Motion of a Frictionless Pendulum 522
11.11.2 Compliance of Arterial Walls . . . . . . . . . . . . 523
11.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
12. Two-Point Boundary Value Problems 531
12.1 A Model Problem . . . . . . . . . . . . . . . . . . . . . . . 531
12.2 Finite Difference Approximation . . . . . . . . . . . . . . . 533Contents xix
12.2.1 Stability Analysis by the Energy Method . . . . . 534
12.2.2 Convergence Analysis . . . . . . . . . . . . . . . . 538
12.2.3 Finite Differences for Two-Point Boundary
Value Problems with Variable Coefficients . . . . . 540
12.3 The Spectral Collocation Method . . . . . . . . . . . . . . 542
12.4 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . 544
12.4.1 Integral Formulation of Boundary-Value Problems 544
12.4.2 A Quick Introduction to Distributions . . . . . . . 546
12.4.3 Formulation and Properties of the
Galerkin Method . . . . . . . . . . . . . . . . . . . 547
12.4.4 Analysis of the Galerkin Method . . . . . . . . . . 548
12.4.5 The Finite Element Method . . . . . . . . . . . . . 550
12.4.6 Implementation Issues . . . . . . . . . . . . . . . . 556
12.4.7 Spectral Methods . . . . . . . . . . . . . . . . . . . 559
12.5 Advection-Diffusion Equations . . . . . . . . . . . . . . . . 560
12.5.1 Galerkin Finite Element Approximation . . . . . . 561
12.5.2 The Relationship Between Finite Elements and
Finite Differences; the Numerical Viscosity . . . . 563
12.5.3 Stabilized Finite Element Methods . . . . . . . . . 567
12.6 A Quick Glance to the Two-Dimensional Case . . . . . . . 572
12.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 575
12.7.1 Lubrication of a Slider . . . . . . . . . . . . . . . . 575
12.7.2 Vertical Distribution of Spore
Concentration over Wide Regions . . . . . . . . . . 576
12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
13. Parabolic and Hyperbolic Initial Boundary
Value Problems 581
13.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . 581
13.2 Finite Difference Approximation of the Heat Equation . . 584
13.3 Finite Element Approximation of the Heat Equation . . . 586
13.3.1 Stability Analysis of the θ-Method . . . . . . . . . 588
13.4 Space-Time Finite Element Methods for the
Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 593
13.5 Hyperbolic Equations: A Scalar Transport Problem . . . . 597
13.6 Systems of Linear Hyperbolic Equations . . . . . . . . . . 599
13.6.1 The Wave Equation . . . . . . . . . . . . . . . . . 601
13.7 The Finite Difference Method for Hyperbolic Equations . . 602
13.7.1 Discretization of the Scalar Equation . . . . . . . . 602
13.8 Analysis of Finite Difference Methods . . . . . . . . . . . . 605
13.8.1 Consistency . . . . . . . . . . . . . . . . . . . . . . 605
13.8.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . 605
13.8.3 The CFL Condition . . . . . . . . . . . . . . . . . 606
13.8.4 Von Neumann Stability Analysis . . . . . . . . . . 608
13.9 Dissipation and Dispersion . . . . . . . . . . . . . . . . . . 611xx Contents
13.9.1 Equivalent Equations . . . . . . . . . . . . . . . . 614
13.10 Finite Element Approximation of Hyperbolic Equations . . 618
13.10.1 Space Discretization with Continuous and
Discontinuous Finite Elements . . . . . . . . . . . 618
13.10.2 Time Discretization . . . . . . . . . . . . . . . . . 620
13.11 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 623
13.11.1 Heat Conduction in a Bar . . . . . . . . . . . . . . 623
13.11.2 A Hyperbolic Model for Blood Flow
Interaction with Arterial Walls . . . . . . . . . . . 623
13.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
References 627
Index of MATLAB Programs 643
Index 647
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