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 |  موضوع: كتاب Finite Element Methods and Their Applications - Zhangxin Chen السبت 16 أغسطس 2014, 12:09 pm | |
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Finite Element Methods and Their Applications - Zhangxin Chen
ويتناول الموضوعات الأتية :
Contents
1 Elementary Finite Elements 1
11 Introduction 2
111 A One-Dimensional Model Problem 2
112 A Two-Dimensional Model Problem 9
113 An Extension to General Boundary Conditions14
114 Programming Considerations 16
12 Sobolev Spaces 19
121 LebesgueSpaces 20
122 Weak Derivatives 21
123 Sobolev Spaces 22
124 Poincar´e’s Inequality 23
125 Duality and Negative Norms 25
13 Abstract Variational Formulation 26
131 An Abstract Formulation 26
132 The Finite Element Method28
133 Examples 30
14 Finite Element Spaces 35
141 Triangles 35
142 Rectangles 40
143 Three Dimensions 42
144 AC
Element 44
15 GeneralDomains 46
16 Quadrature Rules 49
17 Finite Elements for Transient Problems 50
171 A One-Dimensional Model Problem 51
172 ASemi-Discrete Scheme in Space 52
173 Fully Discrete Schemes 55
18 Finite Elements for Nonlinear Problems58
181 Linearization Approaches 59
182 Implicit Time Approximations 60
183 Explicit Time Approximations 61
19 Approximation Theory 62
191 Interpolation Errors 62
192 Error Estimates for Elliptic Problems67
X Contents
-Error Estimates 68
110 Linear SystemSolution Techniques 70
1101 Gaussian Elimination 70
1102 The Conjugate Gradient Algorithm 76
111 BibliographicalRemarks 81
112 Exercises 81
2 Nonconforming Finite Elements 87
21 Second-Order Problems 87
211 Nonconforming Finite Elements on Triangles 89
212 Nonconforming Finite Elements on Rectangles92
213 Nonconforming Finite Elements on Tetrahedra95
214 Nonconforming Finite Elements on Parallelepipeds95
215 Nonconforming Finite Elements on Prisms97
22 Fourth-Order Problems 98
221 The Morley Element 100
222 The Fraeijs de Veubeke Element 102
223 The Zienkiewicz Element 103
224 The Adini Element104
23 Nonlinear Problems 105
24 TheoreticalConsiderations 106
241 An Abstract Formulation 106
242 Applications 109
25 BibliographicalRemarks 113
26 Exercises 113
3 Mixed Finite Elements 117
31 A One-Dimensional Model Problem118
32 ATwo-DimensionalModelProblem 123
33 Extension to Boundary Conditions of Other Types 126
331 A Neumann Boundary Condition126
332 A Boundary Condition of Third Type128
34 Mixed Finite Element Spaces 128
341 Mixed Finite Element Spaces on Triangles130
342 Mixed Finite Element Spaces on Rectangles 133
343 Mixed Finite Element Spaces on Tetrahedra 136
344 Mixed Finite Element Spaces on Parallelepipeds 137
345 Mixed Finite Element Spaces on Prisms 140
35 Approximation Properties143
36 Mixed Methods for Nonlinear Problems143
37 Linear SystemSolution Techniques 145
371 Introduction 145
372 The Uzawa Algorithm 146
373 The Minimum Residual Iterative Algorithm 147
374 Alternating Direction Iterative Algorithms148
Contents XI
375 Mixed-Hybrid Algorithms 150
376 An Equivalence Relationship 152
38 TheoreticalConsiderations 154
381 An Abstract Formulation 154
382 The Mixed Finite Element Method 158
383 Examples 161
384 Construction of Projection Operators162
385 Error Estimates 164
39 BibliographicalRemarks 166
310 Exercises 167
4 Discontinuous Finite Elements173
41 Advection Problems 173
411 DGMethods 174
412 Stabilized DGMethods 178
42 Diffusion Problems 183
421 Symmetric DG Method186
422 Symmetric Interior Penalty DG Method 187
423 Non-Symmetric DG Method188
424 Non-Symmetric Interior Penalty DG Method 189
425 Remarks 192
43 Mixed Discontinuous Finite Elements 194
431 AOne-DimensionalProblem 194
432 Multi-Dimensional Problems203
433 Nonlinear Problems206
44 TheoreticalConsiderations 208
441 DGMethods 208
442 Stabilized DGMethods 210
45 BibliographicalRemarks 212
46 Exercises 212
5 Characteristic Finite Elements215
51 An Example 216
52 The Modified Method of Characteristics 218
521 A One-Dimensional Model Problem 218
522 Periodic Boundary Conditions 222
523 Extension to Multi-Dimensional Problems222
524 Discussion of a Conservation Relation224
53 The Eulerian-Lagrangian Localized Adjoint Method 226
531 A One-Dimensional Model Problem 226
532 Extension to Multi-Dimensional Problems236
54 The Characteristic MixedMethod 242
55 The Eulerian-Lagrangian Mixed Discontinuous Method 245
56 Nonlinear Problems 248
57 Remarks on Characteristic Finite Elements 250
XII Contents
58 TheoreticalConsiderations 250
59 BibliographicalRemarks 258
510 Exercises 258
6 Adaptive Finite Elements261
61 LocalGridRefinement inSpace 262
611 Regular H-Schemes 263
612 Irregular H-Schemes 265
613 Unrefinements 266
62 Data Structures 267
63 A-Posteriori Error Estimates for Stationary Problems270
631 ResidualEstimators 271
632 Local Problem-Based Estimators277
633 Averaging-Based Estimators281
634 HierarchicalBasis Estimators 283
635 Efficiency of Error Estimators 287
64 A-Posteriori Error Estimates for Transient Problems 289
65 A-Posteriori Error Estimates for Nonlinear Problems 292
66 TheoreticalConsiderations 293
661 An Abstract Theory 294
662 Applications 297
67 BibliographicalRemarks 302
68 Exercises 302
7 Solid Mechanics305
71 Introduction 305
711 Kinematics 305
712 Equilibrium 306
713 MaterialLaws 306
72 VariationalFormulations 308
721 The Displacement Form 308
722 The Mixed Form 309
73 Finite Element Methods 310
731 Finite Elements and Locking Effects 310
732 Mixed Finite Elements 311
733 Nonconforming Finite Elements 313
74 TheoreticalConsiderations 314
75 BibliographicalRemarks 319
76 Exercises 319
8 Fluid Mechanics321
81 Introduction 321
82 VariationalFormulations 323
821 The Galerkin Approach323
822 The Mixed Formulation 324
Contents XIII
83 Finite Element Methods 324
831 Galerkin Finite Elements 324
832 Mixed Finite Elements 325
833 Nonconforming Finite Elements 326
84 The Navier-Stokes Equation 329
85 TheoreticalConsiderations 330
86 BibliographicalRemarks 333
87 Exercises 333
9 Fluid Flow in Porous Media337
91 Two-Phase Immiscible Flow 338
911 The Phase Formulation 340
912 The Weighted Formulation 342
913 The GlobalFormulation 342
92 Mixed Finite Elements for Pressure 343
93 Characteristic Methods for Saturation 345
94 ANumericalExample 346
95 TheoreticalConsiderations 349
951 Analysis for the Pressure Equation 349
952 Analysis for the Saturation Equation 351
96 BibliographicalRemarks 361
97 Exercises 362
10 Semiconductor Modeling363
101 Three Semiconductor Models 364
1011 The Drift-Diffusion Model 364
1012 The Hydrodynamic Model 366
1013 The QuantumHydrodynamic Model 367
102 NumericalMethods 368
1021 The Drift-Diffusion Model 368
1022 The Hydrodynamic Model 371
1023 The QuantumHydrodynamic Model 378
103 ANumericalExample 379
104 BibliographicalRemarks 384
105 Exercises 384
A Nomenclature385
References391
Index
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