The finite element method : theory, implementation, and applications

larson......Page 1
bfm%3A978-3-642-33287-6%2F1......Page 2
The Finite Element Method: Theory, Implementation, and Applications......Page 4
Preface......Page 6
Acknowledgements......Page 8
Contents......Page 10
1.1.1 The Space of Linear Polynomials......Page 19
1.1.2 The Space of Continuous Piecewise LinearPolynomials......Page 20
1.2.1 Linear Interpolation......Page 22
1.2.2 Continuous Piecewise Linear Interpolation......Page 25
1.3.1 Definition......Page 26
1.3.2 Derivation of a Linear System of Equations......Page 28
1.3.4 A Priori Error Estimate......Page 29
1.4.2 The Trapezoidal Rule......Page 31
1.4.3 Simpson's Formula......Page 32
1.5.1 Assembly of the Mass Matrix......Page 33
1.5.2 Assembly of the Load Vector......Page 36
1.6 Problems......Page 38
2.1.1 A Two-point Boundary Value Problem......Page 41
2.1.3 Finite Element Approximation......Page 42
2.1.4 Derivation of a Linear System of Equations......Page 43
2.1.5 Basic Algorithm to Compute the Finite Element Solution......Page 44
2.1.6 A Priori Error Estimate......Page 45
2.2 Mathematical Modeling......Page 46
2.2.2 Boundary Conditions for the Heat Equation......Page 47
2.2.3 Derivation of a Differential Equation for the Deformation of a Bar......Page 48
2.2.4.3 Robin Boundary Conditions......Page 49
2.3.1 Variational Formulation......Page 50
2.4.1 Assembly of the Stiffness Matrix and Load Vector......Page 51
2.4.2 A Finite Element Solver for a General Two-Point Boundary Value Problem......Page 55
2.5.1 A Posteriori Error Estimate......Page 56
2.5.2 Adaptive Mesh Refinement......Page 58
2.7 Problems......Page 60
3.1.1 Triangulations......Page 63
3.1.2 Data Structures for a Triangulation......Page 64
3.1.3 Mesh Generation......Page 65
3.2.1 The Space of Linear Polynomials......Page 67
3.2.2 The Space of Continuous Piecewise LinearPolynomials......Page 68
3.2.3 The Space of Piecewise Constants......Page 70
3.3.1 Linear Interpolation......Page 71
3.3.2 Continuous Piecewise Linear Interpolation......Page 73
3.4.2 Derivation of a Linear System of Equations......Page 74
3.4.4 Existence and Uniqueness of the L2-Projection......Page 75
3.4.5 A Priori Error Estimate......Page 76
3.5 Quadrature and Numerical Integration......Page 77
3.6.1 Assembly of the Mass Matrix......Page 78
3.6.2 Assembly of the Load Vector......Page 82
3.6.3 Properties of the Mass Matrix......Page 84
3.7 Further Reading......Page 85
3.8 Problems......Page 86
4.1 Green's Formula......Page 88
4.2.2 Variational Formulation......Page 89
4.2.4 Derivation of a Linear System of Equations......Page 90
4.2.5 Basic Algorithm to Compute the Finite Element Solution......Page 91
4.3 Some Useful Inequalities......Page 92
4.4.1 Existence and Uniqueness of the Finite Element Solution......Page 94
4.4.2 A Priori Error Estimates......Page 95
4.4.3 Properties of the Stiffness Matrix......Page 98
4.6 Computer Implementation......Page 100
4.6.1 Assembly of the Stiffness Matrix......Page 101
4.6.2 Assembling the Boundary Conditions......Page 103
4.6.3.1 Potential Flow Over a Wing......Page 105
4.7 The Dirichlet Problem......Page 108
4.8 The Neumann Problem......Page 110
4.9 The Eigenvalue Problem......Page 112
4.10.1 A Posteriori Error Estimate......Page 114
4.10.2 Adaptive Mesh Refinement......Page 116
4.10.2.1 Rivara Refinement......Page 117
4.10.2.2 Regular Refinement......Page 118
4.10.3 Adaptive Finite Elements Using MATLAB......Page 119
4.10.4 Duality Based a Posteriori Error Estimate......Page 122
4.12 Problems......Page 126
5.1 Finite Difference Methods for Systems of ODE......Page 129
5.2.1 Derivation of the Transient Heat Equation......Page 132
5.2.2 Model Problem......Page 133
5.2.4 Spatial Discretization......Page 134
5.2.5 Time Discretization......Page 136
5.3.1 A Space Discrete Estimate......Page 137
5.4 A Priori Error Estimates......Page 138
5.4.2 A Space Discrete Estimate......Page 139
5.4.3 A Fully Discrete Estimate......Page 140
5.5 Computer Implementation......Page 142
5.5.1 Mass Lumping......Page 143
5.6 Space-Time Finite Element Approximation......Page 145
5.6.1 A Posteriori Error Estimate......Page 147
5.7.1 Derivation of the Acoustic Wave Equation......Page 150
5.7.2 Model Problem......Page 151
5.7.4 Spatial Discretization......Page 152
5.7.6 Conservation of Energy......Page 153
5.8 Computer Implementation......Page 155
5.9 Further Reading......Page 156
5.10 Problems......Page 157
6.1 Linear Systems......Page 159
6.2.1 Triangular Systems, Backward and Forward Substitution......Page 160
6.2.2 Fill-ins, Graphs, and Symbolic and Numeric Factorization......Page 161
6.2.3.1 Minimum Degree......Page 166
6.2.3.2 Nested Dissection......Page 167
6.3.1 Basic Iterative Methods......Page 169
6.3.4 Convergence Analysis......Page 171
6.3.5 Projection Methods......Page 173
6.3.6 One-Dimensional Projection Methods......Page 174
6.3.7 Krylov Subspaces......Page 175
6.3.8 The Conjugate Gradient Method......Page 178
6.3.9 The Generalized Minimal Residual Method......Page 181
6.4 Preconditioning......Page 182
6.4.2 Polynomial Preconditioners......Page 183
6.4.3 Incomplete Factorizations......Page 184
6.5.1 The Smoothing Property......Page 185
6.5.2 A Two-Grid Method......Page 186
6.6 Further Reading......Page 189
6.7 Problems......Page 190
7.1.1 Normed and Inner Product Vector Spaces......Page 193
7.1.2 Banach and Hilbert Spaces......Page 196
7.1.4 Weak Derivatives......Page 197
7.1.5 Sobolev Spaces......Page 199
7.1.6 Traces......Page 201
7.2.1 The Clement Interpolant......Page 202
7.2.2 The Scott-Zhang Interpolant......Page 203
7.3.2 The Riesz Representation Theorem......Page 204
7.3.3 An Equivalent Minimization Problem......Page 205
7.3.4 The Lax-Milgram Lemma......Page 207
7.3.5.1 Poisson's Equation......Page 208
7.3.5.2 The Diffusion-Reaction Equation......Page 209
7.3.6 A General Linear Second Order Elliptic Problem......Page 210
7.4.2 The Stiffness Matrix and Load Vector......Page 212
7.4.4 A Priori Error Estimates......Page 213
7.4.5 A Posteriori Error Estimates......Page 214
7.6 Problems......Page 216
8.1.1 Formal Definition of a Finite Element......Page 218
8.1.2 Shape Functions for the Linear Lagrange Triangle......Page 220
8.1.3 Shape Functions for the Quadratic LagrangeTriangle......Page 221
8.1.5 Shape Functions for the Bilinear Quadrilateral Element......Page 223
8.1.6 Shape Functions for the Linear LagrangeTetrahedron......Page 224
8.1.7 Shape Functions for the Trilinear Brick Element......Page 225
8.2 The Isoparametric Mapping......Page 226
8.2.1 Numerical Quadrature Revisited......Page 228
8.2.2 Renumbering the Mesh for Quadratic Nodes......Page 230
8.3 Some Exotic Finite Elements......Page 232
8.3.2 The Crouzeix-Raviart Element......Page 233
8.3.3 The Lowest Order Raviart-Thomas Element......Page 235
8.3.4 The Lowest Order Nédélec Element......Page 236
8.5 Problems......Page 237
9.1 Piccard Iteration......Page 239
9.2 Newton's Method......Page 240
9.3 The Non-linear Poisson Equation......Page 241
9.3.1 The Newton-Galerkin Method......Page 242
9.3.2 Finite Element Approximation......Page 243
9.3.3 Computer Implementation......Page 244
9.3.4 Piccard Iteration as a Simplified Newton Method......Page 246
9.4.2 Space Discretization......Page 247
9.4.4 Piccard Iteration......Page 248
9.4.5 Newton's Method......Page 249
9.5 Numerical Approximations of the Jacobian......Page 251
9.5.2 Broyden's Method......Page 252
9.7 Problems......Page 253
10.1 The Transport Equation......Page 254
10.1.2 Standard Galerkin Finite Element Approximation......Page 255
10.1.3 Computer Implementation......Page 256
10.1.4 The Need for Stabilization......Page 257
10.1.5 Least-Squares Stabilization......Page 259
10.1.6 GLS Finite Element Approximation......Page 261
10.1.7 A Priori Error Estimate......Page 262
10.1.8.1 Heat Transfer in a Fluid Flow......Page 264
10.3 Problems......Page 269
11.1.1 Cauchy's Equilibrium Equation......Page 270
11.1.2 Constitutive Equations and Hooke's Law......Page 271
11.1.3 Boundary Conditions......Page 272
11.2.1 Weak Form......Page 273
11.2.2 Existence and Uniqueness of the Solution......Page 275
11.2.3 Finite Element Approximation......Page 276
11.2.4 A Priori Error Estimate......Page 277
11.2.5 Engineering Notation......Page 278
11.2.6 Computer Implementation......Page 280
11.2.6.1 Verifying the Energy Norm Convergence......Page 285
11.2.7 A Posteriori Error Estimate......Page 287
11.3 Linear Thermoelasticity......Page 290
11.4 Linear Elastodynamics......Page 292
11.4.1.1 Eigenvalues and Eigenmodes of a Steel Bracket......Page 293
11.4.2.1 Rayleigh Damping......Page 295
11.4.2.2 Newmark's Method......Page 296
11.5.1 Governing Equations......Page 297
11.5.1.2 The C1 Continuity Requirement and the Morley Approximation......Page 298
11.7 Problems......Page 299
12.1.1 Conservation of Mass......Page 301
12.1.2 Momentum Balance......Page 302
12.1.3 Incompressible Newtonian Fluids......Page 303
12.2.1 The Stationary Stokes System......Page 304
12.2.2 Weak Form......Page 305
12.2.3 Equivalent Optimization Problem......Page 307
12.2.4 The Continuous Inf-Sup Condition......Page 308
12.2.6 The Discrete Inf-Sup Condition......Page 310
12.2.6.1 The Uzawa Algorithm......Page 312
12.2.7.1 The Taylor-Hood Element......Page 313
12.2.7.3 The Non-conforming P1-P0 Element......Page 314
12.2.8.1 Fortin's Trick......Page 315
12.2.9 A Priori Error Estimate......Page 317
12.2.10.1 The Lid-Driven Cavity......Page 319
12.2.11 A Posteriori Error Estimate......Page 324
12.2.12.2 Galerkin Least Squares Stabilization......Page 326
12.3 The Navier-Stokes Equations......Page 328
12.3.1.1 Time Discretization......Page 329
12.3.1.2 Space Discretization......Page 331
12.3.2.1 The DFG Benchmark......Page 332
12.4 Further Reading......Page 335
12.5 Problems......Page 336
13.1 Governing Equations......Page 338
13.1.2 Gauss' Laws......Page 339
13.1.3 Ampère's Law......Page 340
13.1.5 Maxwell's Equations......Page 341
13.1.6 Initial- and Boundary Conditions......Page 342
13.2 The Electric and Magnetic Wave Equations......Page 343
13.4 Electro- and Magnetostatics......Page 344
13.5.1 Boundary Conditions......Page 345
13.5.3 Weak Form......Page 346
13.5.4 Finite Element Approximation......Page 347
13.5.5 Computer Implementation......Page 349
13.5.5.1 Electromagnetic Scattering from a Cylinder......Page 351
13.6 The Magnetostatic Potential Equation......Page 355
13.6.2 Weak Form......Page 357
13.6.3 The Helmholtz Decomposition......Page 358
13.6.4 Existence and Uniqueness of the Solution......Page 360
13.6.5 Finite Element Approximation......Page 361
13.6.6 A Priori Error Estimate......Page 362
13.6.7 A Posteriori Error Estimate......Page 363
13.8 Problems......Page 365
14.1.1 Model Problem......Page 366
14.1.2 Discontinuous Finite Element Spaces......Page 367
14.1.3 The Discontinuous Galerkin Method......Page 368
14.1.4 Stability Estimates......Page 369
14.1.5 A Priori Error Estimates......Page 372
14.2.2 The Symmetric Interior Penalty Method......Page 373
14.2.3 Approximation Properties......Page 374
14.2.4 A Priori Error Estimates......Page 375
14.2.5 Non-symmetric Versions of the Method......Page 377
14.2.6 Computer Implementation......Page 378
14.3 The Transport Equation with Diffusion......Page 381
14.5 Problems......Page 383
A.1 Tri2Edge.m......Page 384
A.3 Tri2Tri.m......Page 385
A.5 Airfoilg.m......Page 386
A.7 DFGg.m......Page 387
A.8 Scatterg.m......Page 388
References......Page 389
Index......Page 392