Finite element methods and their applications

Zhangxin Chen9783540240785, 3540240780

This book serves as a text for one- or two-semester courses for upper-level undergraduates and beginning graduate students and as a professional reference for people who want to solve partial differential equations (PDEs) using finite element methods. The author has attempted to introduce every concept in the simplest possible setting and maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Quite a lot of attention is given to discontinuous finite elements, characteristic finite elements, and to the applications in fluid and solid mechanics including applications to porous media flow, and applications to semiconductor modeling. An extensive set of exercises and references in each chapter are provided.

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Table of contents :
Front cover ……Page 1
Scientific Computation Series ……Page 2
Title page ……Page 3
Date-line ……Page 4
Preface ……Page 5
Contents ……Page 9
1 Elementary Finite Elements ……Page 15
1.1.1 A One-Dimensional Model Problem ……Page 16
1.1.2 A Two-Dimensional Model Problem ……Page 23
1.1.3 An Extension to General Boundary Conditions ……Page 28
1.1.4 Programming Considerations ……Page 30
1.2 Sobolev Spaces ……Page 33
1.2.1 Lebesgue Spaces ……Page 34
1.2.2 Weak Derivatives ……Page 35
1.2.3 Sobolev Spaces ……Page 36
1.2.4 Poincare’s Inequality ……Page 37
1.2.5 Duality and Negative Norms ……Page 39
1.3.1 An Abstract Formulation ……Page 40
1.3.2 The Finite Element Method ……Page 42
1.3.3 Examples ……Page 44
1.4.1 Triangles ……Page 49
1.4.2 Rectangles ……Page 54
1.4.3 Three Dimensions ……Page 56
1.4.4 A $C^1$ Element ……Page 58
1.5 General Domains ……Page 60
1.6 Quadrature Rules ……Page 63
1.7 Finite Elements for Transient Problems ……Page 64
1.7.1 A One-Dimensional Model Problem ……Page 65
1.7.2 A Semi-Discrete Scheme in Space ……Page 66
1.7.3 Fully Discrete Schemes ……Page 69
1.8 Finite Elements for Nonlinear Problems ……Page 72
1.8.1 Linearization Approaches ……Page 73
1.8.2 Implicit Time Approximations ……Page 74
1.8.3 Explicit Time Approximations ……Page 75
1.9.1 Interpolation Errors ……Page 76
1.9.2 Error Estimates for Elliptic Problems ……Page 81
1.9.3 $L^2$-Error Estimates ……Page 82
1.10.1 Gaussian Elimination ……Page 84
1.10.2 The Conjugate Gradient Algorithm ……Page 90
1.12 Exercises ……Page 95
2.1 Second-Order Problems ……Page 101
2.1.1 Nonconforming Finite Elements on Triangles ……Page 103
2.1.2 Nonconforming Finite Elements on Rectangles ……Page 106
2.1.4 Nonconforming Finite Elements on Parallelepipeds ……Page 109
2.1.5 Nonconforming Finite Elements on Prisms ……Page 111
2.2 Fourth-Order Problems ……Page 112
2.2.1 The Morley Element ……Page 114
2.2.2 The Fraeijs de Veubeke Element ……Page 116
2.2.3 The Zienkiewicz Element ……Page 117
2.2.4 The Adini Element ……Page 118
2.3 Nonlinear Problems ……Page 119
2.4.1 An Abstract Formulation ……Page 120
2.4.2 Applications ……Page 123
2.6 Exercises ……Page 127
3 Mixed Finite Elements ……Page 131
3.1 A One-Dimensional Model Problem ……Page 132
3.2 A Two-Dimensional Model Problem ……Page 137
3.3.1 A Neumann Boundary Condition ……Page 140
3.4 Mixed Finite Element Spaces ……Page 142
3.4.1 Mixed Finite Element Spaces on Triangles ……Page 144
3.4.2 Mixed Finite Element Spaces on Rectangles ……Page 147
3.4.3 Mixed Finite Element Spaces on Tetrahedra ……Page 150
3.4.4 Mixed Finite Element Spaces on Parallelepipeds ……Page 151
3.4.5 Mixed Finite Element Spaces on Prisms ……Page 154
3.6 Mixed Methods for Nonlinear Problems ……Page 157
3.7.1 Introduction ……Page 159
3.7.2 The Uzawa Algorithm ……Page 160
3.7.3 The Minimum Residual Iterative Algorithm ……Page 161
3.7.4 Alternating Direction Iterative Algorithms ……Page 162
3.7.5 Mixed-Hybrid Algorithms ……Page 164
3.7.6 An Equivalence Relationship ……Page 166
3.8.1 An Abstract Formulation ……Page 168
3.8.2 The Mixed Finite Element Method ……Page 172
3.8.3 Examples ……Page 175
3.8.4 Construction of Projection Operators ……Page 176
3.8.5 Error Estimates ……Page 178
3.9 Bibliographical Remarks ……Page 180
3.10 Exercises ……Page 181
4.1 Advection Problems ……Page 187
4.1.1 DG Methods ……Page 188
4.1.2 Stabilized DG Methods ……Page 192
4.2 Diffusion Problems ……Page 197
4.2.1 Symmetric DG Method ……Page 200
4.2.2 Symmetric Interior Penalty DG Method ……Page 201
4.2.3 Non-Symmetric DG Method ……Page 202
4.2.4 Non-Symmetric Interior Penalty DG Method ……Page 203
4.2.5 Remarks ……Page 206
4.3.1 A One-Dimensional Problem ……Page 208
4.3.2 Multi-Dimensional Problems ……Page 217
4.3.3 Nonlinear Problems ……Page 220
4.4.1 DG Methods ……Page 222
4.4.2 Stabilized DG Methods ……Page 224
4.6 Exercises ……Page 226
5 Characteristic Finite Elements ……Page 229
5.1 An Example ……Page 230
5.2.1 A One-Dimensional Model Problem ……Page 232
5.2.3 Extension to Multi-Dimensional Problems ……Page 236
5.2.4 Discussion of a Conservation Relation ……Page 238
5.3.1 A One-Dimensional Model Problem ……Page 240
5.3.2 Extension to Multi-Dimensional Problems ……Page 250
5.4 The Characteristic Mixed Method ……Page 256
5.5 The Eulerian-Lagrangian Mixed Discontinuous Method ……Page 259
5.6 Nonlinear Problems ……Page 262
5.8 Theoretical Considerations ……Page 264
5.10 Exercises ……Page 272
6 Adaptive Finite Elements ……Page 275
6.1 Local Grid Refinement in Space ……Page 276
6.1.1 Regular H-Schemes ……Page 277
6.1.2 Irregular H-Schemes ……Page 279
6.1.3 Unrefinements ……Page 280
6.2 Data Structures ……Page 281
6.3 A-Posteriori Error Estimates for Stationary Problems ……Page 284
6.3.1 Residual Estimators ……Page 285
6.3.2 Local Problem-Based Estimators ……Page 291
6.3.3 Averaging-Based Estimators ……Page 295
6.3.4 Hierarchical Basis Estimators ……Page 297
6.3.5 Efficiency of Error Estimators ……Page 301
6.4 A-Posteriori Error Estimates for Transient Problems ……Page 303
6.5 A-Posteriori Error Estimates for Nonlinear Problems ……Page 306
6.6 Theoretical Considerations ……Page 307
6.6.1 An Abstract Theory ……Page 308
6.6.2 Applications ……Page 311
6.8 Exercises ……Page 316
7.1.1 Kinematics ……Page 319
7.1.3 Material Laws ……Page 320
7.2.1 The Displacement Form ……Page 322
7.2.2 The Mixed Form ……Page 323
7.3.1 Finite Elements and Locking Effects ……Page 324
7.3.2 Mixed Finite Elements ……Page 325
7.3.3 Nonconforming Finite Elements ……Page 327
7.4 Theoretical Considerations ……Page 328
7.6 Exercises ……Page 333
8.1 Introduction ……Page 335
8.2.1 The Galerkin Approach ……Page 337
8.3.1 Galerkin Finite Elements ……Page 338
8.3.2 Mixed Finite Elements ……Page 339
8.3.3 Nonconforming Finite Elements ……Page 340
8.4 The Navier-Stokes Equation ……Page 343
8.5 Theoretical Considerations ……Page 344
8.7 Exercises ……Page 347
9 Fluid Flow in Porous Media ……Page 351
9.1 Two-Phase Immiscible Flow ……Page 352
9.1.1 The Phase Formulation ……Page 354
9.1.3 The Global Formulation ……Page 356
9.2 Mixed Finite Elements for Pressure ……Page 357
9.3 Characteristic Methods for Saturation ……Page 359
9.4 A Numerical Example ……Page 360
9.5.1 Analysis for the Pressure Equation ……Page 363
9.5.2 Analysis for the Saturation Equation ……Page 365
9.6 Bibliographical Remarks ……Page 375
9.7 Exercises ……Page 376
10 Semiconductor Modeling ……Page 377
10.1.1 The Drift-Diffusion Model ……Page 378
10.1.2 The Hydrodynamic Model ……Page 380
10.1.3 The Quantum Hydrodynamic Model ……Page 381
10.2.1 The Drift-Diffusion Model ……Page 382
10.2.2 The Hydrodynamic Model ……Page 385
10.2.3 The Quantum Hydrodynamic Model ……Page 392
10.3 A Numerical Example ……Page 393
10.5 Exercises ……Page 398
A Nomenclature ……Page 399
References ……Page 405
Index ……Page 419
Back cover ……Page 425