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Tables of Contents for Generalized Difference Methods for Differential Equations
Chapter/Section Title
Page #
Page Count
Preface
iii
 
Preliminaries
1
44
Sobolev Spaces
1
18
Smooth approximations. Fundamental lemma of variational methods
1
2
Generalized derivatives and Sobolev spaces
3
4
Imbedding and trace theorems
7
2
Finite element spaces
9
6
Interpolation error estimates in Sobolev spaces
15
4
Variational Problems and Their Approximations
19
26
Abstract variational from
19
4
Green's Formulas and variational problems
23
4
Well-posedness of variational problems
27
2
Approximation methods. A necessary and sufficient condition for approximate-solvability
29
6
Galerkin methods
35
3
Generalized Galerkin methods
38
5
Bibliography and Comments
43
2
Two Point Boundary Value Problems
45
190
Basic Ideas of the Generalized Difference Method
45
8
A Variational From
45
2
Galerkin methods
47
1
Generalized Galerkin Variational principles
48
4
Generalized difference methods
52
1
Linear Element Difference Schemes
53
8
Trial and test function spaces
53
3
Difference equations
56
1
Convergence estimates
57
4
Quadratic Element Difference Schemes
61
11
Trial and test spaces
61
2
Difference equations
63
3
Convergence order estimates
66
6
Cubic Element Difference Schemes
72
16
Trial and test spaces
73
3
Generalized difference methods
76
3
Some lemmas
79
5
Existence, uniqueness and stability
84
2
Convergence order estimates
86
1
Numerical examples
87
1
Estimates in L2 and Maximum Norms
88
3
L2-estimates
88
3
Maximum norm estimates
91
1
Superconvergence
91
10
Optimal stress points
92
4
Superconvergence for linear element difference schemes
96
1
Superconvergence for cubic element difference schemes
97
4
Generalized Difference Methods for a Fourth Order Equation
101
10
Generalized difference equations
101
3
Positive definiteness of a(uh, &Pihuh)
104
2
Convergence order estimates
106
2
Numerical Examples
108
1
Bibliography and Comments
108
3
Second Order Elliptic Equations
Introduction
111
3
Generalized Difference Methods on Triangular Meshes
114
17
Trial and test function spaces
114
5
Generalized difference equation
119
5
a priori estimates
124
5
Error estimates
129
2
Generalized Difference Methods on Quadrilateral Meshes
131
8
Trial and test function spaces
131
3
Generalized difference equation
134
3
Convergence order estimates
137
2
Quadratic Element Difference Schemes
139
15
Trial and test function spaces
139
2
Generalized difference equation
141
4
a priori estimates
145
5
Error estimates
150
2
Numerical example
152
2
Cubic Element Difference Schemes
154
13
Trial and test function spaces
154
2
Generalized difference equation
156
2
a priori estimates
158
6
Error estimates
164
3
L2 and Maximum North Estimates
167
7
L2 Estimates
167
6
A maximum estimate and some remarks
173
1
Superconvergences
174
13
Week estimate of interpolations
174
7
Superconvergence estimates
181
1
Bibliography and comments
182
5
Fourth Order and Nonlinear Elliptic Equations
Mixed generalized Difference Methods Based on Ciarlet-Raviart Variational Principle
187
9
Mixed generalized Difference equations
188
5
Error Estimates
193
3
Mixed Generalized Difference Methods Based on Hermann-Miyoshi Variational Principle
196
8
Mixed generalized difference equations
197
1
Numerical experiments
198
6
Nonconforming Generalized Difference Method Based on Zienkiewicz Elements
204
16
Variational principle
202
2
Generalized difference schemes based on Zienkiewicz elements
204
3
Error analyses
207
11
Numerical experiment
218
2
Nonconforming Generalized Difference Methods Based on Adini Elements
220
7
Generalized difference Scheme
220
2
Error Estimate
222
4
Numerical Example
226
1
Second Order Nonlinear Elliptic Equations
227
8
Generalized Difference scheme
227
4
Error Estimate
231
2
Bibliography and Comments
233
2
Parabolic Equations
235
52
Semi-discrete Generalized Difference Schemes
235
9
Problem and schemes
235
1
Some lemmas
235
3
L2-error estimate
238
4
H1-error estimate
242
2
Fully-discrete Generalized Difference Schemes
244
10
Fully-Discrete schemes
244
1
Error Estimates for backward Euler generalized difference schemes
245
5
Error estimates for Crank-Nicolson generalized difference schemes
250
4
Mass Concentration Methods
254
7
Construction of schemes
254
2
Error estimates for fully-discrete schemes
256
2
Error estimates for fully-discrete schemes
258
3
High Order Element Difference Schemes
261
11
Cubic element difference schemes for one dimensional papabolic equations
261
6
Quadratic element difference schemes for two dimensional parabolic equations
267
5
Generalized Difference Methods for Nonlinear Parabolic Equations
272
15
Problem and Schemes
272
4
Some lemmas
276
7
Error estimates
283
2
Bibliography and Comments
285
2
Hyperbolic Equations
287
38
Generalized difference Methods for Second Order Hyperbolic Equations
287
9
Semi-Discrete generalized difference scheme
288
4
Fully-discrete generalized difference scheme
292
4
Generalized Upwind Schemes for First Order Hyperbolic Equations
296
11
Generalized upwind schemes
297
3
Semi-discrete error estimates
300
3
Fully-Discrete error estimates
303
4
Generalized Upwind Schemes for First Order Hyperbolic Systems
307
10
Integral forms
307
2
Generalized upwind difference schemes
309
1
Estimation of a bilinear form
310
3
Some practical difference schemes
313
3
A numerical example
316
1
Finite Volume Methods for Nonlinear Conservative Hyperbolic Equations
317
8
Bibliography and Comments
323
2
Convection-Dominated Diffusion Problems
325
36
One-Dimensional Characteristic Difference Schemes
326
6
Difference methods based on algebraic interpolations
328
2
Upwind difference schemes
330
2
Generalized Upwind Difference Schemes for Steady-state Problems
332
13
Construction of the difference schemes
333
3
Convergence and error estimate
336
3
Extreme value theorem and uniform convergence
339
5
Mass conservation
344
1
Generalized Upwind Difference Schemes for Nonsteady-state Problems
345
6
Construction of difference schemes
346
2
Convergence and error estimate
348
3
Highly Accurate Generalized upwind Schemes
351
6
Construction of the difference schemes
351
4
Convergence and error estimate
355
2
Upwind Schemes for Nonlinear Convection Problems
357
4
Bibliography and Comments
360
1
Applications
361
60
Planar Elastic Problems
361
6
Displacement methods
362
3
Mixed methods
365
2
Computation of Electromagnetic Fields
367
6
Numerical Simulation of Underground Water Pollution
373
12
Generalized difference scheme
374
4
Generalized upwind difference schemes
378
4
Upwind weighted multi-element balancing method
382
3
Stokes Equation
385
9
Nonconforming generalized difference method
386
3
Convergence and error estimate
389
4
A numerical Example
393
1
Coupled Sound-Heat Problems
394
5
Regularized Long Wave Equations
399
10
Semi-discrete generalized difference schemes
399
5
Fully-discrete generalized difference schemes
404
3
Numerical experiments
407
2
Hierarchical Bass Methods
409
12
Hierarchical Basis
410
3
Application to difference equations
413
3
Iteration methods
416
1
Numerical experiments
417
1
Bibliography and Comments
418
3
Bibliography
421
18
Index
439