Generalized difference methods for differential equations: numerical analysis of finite volume methods
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
2000
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| Series: | Pure and applied mathematics
226 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 442 S. graph. Darst. |
| ISBN: | 0824703308 |
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| adam_text | Titel: Generalized difference methods for differential equations
Autor: Li, Ronghua
Jahr: 2000
Contents
PREFACE iii
1 PRELIMINARIES 1
1.1 Sobolev Spaces............................................1
1.1.1 Smooth approximations. Fundamental lemma
of variational methods............................1
1.1.2 Generalized derivatives and Sobolev spaces . . 3
1.1.3 Imbedding and trace theorems..................7
1.1.4 Finite element spaces ............................9
1.1.5 Interpolation error estimates in Sobolev spaces 15
1.2 Variational Problems and Their Approximations ... 19
1.2.1 Abstract variational form........................19
1.2.2 Green s formulas and variational problems ... 23
1.2.3 Well-posedness of variational problems..........27
1.2.4 Approximation methods. A necessary and suf-
ficient condition for approximate-solvability . . 29
1.2.5 Galerkin methods ................................35
1.2.6 Generalized Galerkin methods ..................38
Bibliography and Comments ..................................43
2 TWO POINT BOUNDARY VALUE PROBLEMS 45
2.1 Basic Ideas of the Generalized Difference
Method....................................................45
2.1.1 A variational form................................45
2.1.2 Galerkin methods ................................47
2.1.3 Generalized Galerkin variational principles . . 48
ix
Contents
2.1.4 Generalized difference methods..................52
2.2 Linear Element Difference Schemes......................53
2.2.1 Trial and test function spaces....................53
2.2.2 Difference equations..............................55
2.2.3 Convergence estimates............................57
2.3 Quadratic Element Difference Schemes..................61
2.3.1 Trial and test spaces..............................61
2.3.2 Difference equations..............................63
2.3.3 Convergence order estimates ....................66
2.4 Cubic Element Difference Schemes......................72
2.4.1 TVial and test spaces..............................73
2.4.2 Generalized difference methods..................76
2.4.3 Some lemmas......................................79
2.4.4 Existence, uniqueness and stability..............84
2.4.5 Convergence order estimates ....................86
2.4.6 Numerical examples..............................87
2.5 Estimates in L2 and Maximum Norms..................88
2.5.1 L2-estimates ......................................88
2.5.2 Maximum norm estimates........................91
2.6 Superconvergence..........................................92
2.6.1 Optimal stress points ...................93
2.6.2 Superconvergence for linear element difference
schemes............................................96
2.6.3 Superconvergence for cubic element difference
schemes............................................97
2.7 Generalized Difference Methods for a
Fourth Order Equation..................101
2.7.1 Generalized difference equations........101
2.7.2 Positive defmiteness of a(u/i,II£u/i).......104
2.7.3 Convergence order estimates ..........106
2.7.4 Numerical examples...............108
Bibliography and Comments .................108
SECOND ORDER ELLIPTIC EQUATIONS 111
3.1 Introduction........................HI
3.2 Generalized Difference Methods on Triangular Meshes 114
Contents xi
3.2.1 TYial and test function spaces..........114
3.2.2 Generalized difference equation.........119
3.2.3 a priori estimates................124
3.2.4 Error estimates..................129
3.3 Generalized Difference Methods on
Quadrilateral Meshes...................131
3.3.1 Trial and test function spaces..............131
3.3.2 Generalized difference equation.........134
3.3.3 Convergence order estimates ..........137
3.4 Quadratic Element Difference Schemes.........139
3.4.1 TYial and test function spaces..........139
3.4.2 Generalized difference equation.........141
3.4.3 a priori estimates ................145
3.4.4 Error estimates..................150
3.4.5 Numerical example................152
3.5 Cubic Element Difference Schemes...........154
3.5.1 Trial and test function spaces..........154
3.5.2 Generalized difference equations ........156
3.5.3 a priori estimates ................158
3.5.4 Error estimates..................164
3.6 L2 and Maximum Norm Estimates...........167
3.6.1 L2 estimates ...................167
3.6.2 A maximum estimate and some remarks .... 173
3.7 Superconvergences....................174
3.7.1 Weak estimate of interpolations.........174
3.7.2 Superconvergence estimates ...........181
Bibliography and Comments .................182
4 FOURTH ORDER AND NONLINEAR ELLIPTIC
EQUATIONS 187
4.1 Mixed Generalized Difference Methods
Based on Ciarlet-Raviart Variational
Principle..........................187
4.1.1 Mixed generalized difference equations.....188
4.1.2 Error estimates..................193
xii
Contents
4.2 Mixed Generalized Difference Methods
Based on Hermann-Miyoshi Variational
Principle..........................
4.2.1 Mixed generalized difference equations.....197
4.2.2 Numerical experiments..............198
4.3 Nonconforming Generalized Difference
Method Based on Zienkiewicz Elements........202
4.3.1 Variational principle...............202
4.3.2 Generalized difference schemes based on
Zienkiewicz elements...............204
4.3.3 Error analyses ..................207
4.3.4 Numerical experiment..............218
4.4 Nonconforming Generalized Difference
Methods Based on Adini Elements...........220
4.4.1 Generalized difference scheme..........220
4.4.2 Error estimate..................222
4.4.3 Numerical example................226
4.5 Second Order Nonlinear Elliptic
Equations.........................227
4.5.1 Generalized difference scheme..........227
4.5.2 Error estimate..................231
Bibliography and Comments .................233
5 PARABOLIC EQUATIONS 235
5.1 Semi-discrete Generalized Difference
Schemes..........................235
5.1.1 Problem and schemes..............235
5.1.2 Some lemmas...................238
5.1.3 L2-error estimate.................241
5.1.4 Hl-error estimate ................242
5.2 Fully-discrete Generalized Difference
Schemes..........................244
5.2.1 Fully-discrete schemes..............244
5.2.2 Error estimates for backward Euler generalized
difference schemes................245
Contents xjjj
5.2.3 Error estimates for Crank-Nicolson generalized
difference schemes................250
5.3 Mass Concentration Methods..............254
5.3.1 Construction of schemes.............254
5.3.2 Error estimates for semi-discrete schemes . . . 256
5.3.3 Error estimates for fully-discrete schemes . . . 258
5.4 High Order Element Difference Schemes........261
5.4.1 Cubic element difference schemes for one-
dimensional parabolic equations.........261
5.4.2 Quadratic element difference schemes for two-
dimensional parabolic equations ........267
5.5 Generalized Difference Methods for
Nonlinear Parabolic Equations.............272
5.5.1 Problem and schemes ..............272
5.5.2 Some lemmas...................276
5.5.3 Error estimates..................283
Bibliography and Comments .................285
6 HYPERBOLIC EQUATIONS 287
6.1 Generalized Difference Methods for
Second Order Hyperbolic Equations..........287
6.1.1 Semi-discrete generalized difference scheme . . 288
6.1.2 Fully-discrete generalized difference scheme . . 292
6.2 Generalized Upwind Schemes for
First Order Hyperbolic Equations............296
6.2.1 Generalized upwind schemes ..........297
6.2.2 Semi-discrete error estimates..........300
6.2.3 Fully-discrete error estimates..........303
6.3 Generalized Upwind Schemes for
First Order Hyperbolic Systems.............307
6.3.1 Integral forms...................307
6.3.2 Generalized upwind difference schemes.....309
6.3.3 Estimation of a bilinear form..........310
6.3.4 Some practical difference schemes........313
6.3.5 A numerical example...............316
xiv
Contents
6.4 Finite Volume Methods for Nonlinear
Conservative Hyperbolic Equations...........317
OrtO
Bibliography and Comments .................0£,°
7 CONVECTION-DOMINATED DIFFUSION
PROBLEMS 325
7.1 One-Dimensional Characteristic
Difference Schemes....................326
7.1.1 Difference methods based on algebraic interpo-
lations .......................328
7.1.2 Upwind difference schemes ...........330
7.2 Generalized Upwind Difference Schemes
for Steady-state Problems................332
7.2.1 Construction of the difference schemes.....333
7.2.2 Convergence and error estimate.........336
7.2.3 Extreme value theorem and uniform convergence .. 339
7.2.4 Mass conservation................344
7.3 Generalized Upwind Difference Schemes
for Nonsteady-state Problems..............345
7.3.1 Construction of difference schemes.......346
7.3.2 Convergence and error estimate .........348
7.4 Highly Accurate Generalized
Upwind Schemes.....................351
7.4.1 Construction of the difference schemes.....351
7.4.2 Convergence and error estimate.........355
7.5 Upwind Schemes for Nonlinear Convection Problems . 357
Bibliography and Comments .................360
8 APPLICATIONS 361
8.1 Planar Elastic Problems.................361
8.1.1 Displacement methods..............362
8.1.2 Mixed methods..................365
8.2 Computation of Electromagnetic Fields........367
8.3 Numerical Simulation of Underground
Water Pollution......................373
8.3.1 Generalized difference scheme..........374
8.3.2 Generalized upwind difference schemes.....378
Contents xv
8.3.3 Upwind weighted multi-element balancing
method......................382
8.4 Stokes Equation .....................385
8.4.1 Nonconforming generalized difference method . 386
8.4.2 Convergence and error estimate.........389
8.4.3 A numerical example...............393
8.5 Coupled Sound-Heat Problems.............394
8.6 Regularized Long Wave Equations...........399
8.6.1 Semi-discrete generalized difference schemes . . 399
8.6.2 Fully-discrete generalized difference schemes . . 404
8.6.3 Numerical experiments..............407
8.7 Hierarchical Basis Methods...............409
8.7.1 Hierarchical Basis ................410
8.7.2 Application to difference equations.......413
8.7.3 Iteration methods ................416
8.7.4 Numerical experiments..............417
Bibliography and Comments .................418
Bibliography 421
Index 439
|
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| spelling | Li, Ronghua Verfasser aut Generalized difference methods for differential equations numerical analysis of finite volume methods Ronghua Li ; Zhongying Chen ; Wei Wu New York [u.a.] Dekker 2000 XV, 442 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 226 Differenzenverfahren - Partielle Differentialgleichung Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Differenzenverfahren (DE-588)4134362-1 s DE-604 Chen, Zhongying Verfasser aut Wu, Wei Verfasser aut Pure and applied mathematics 226 (DE-604)BV000001885 226 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009153699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Li, Ronghua Chen, Zhongying Wu, Wei Generalized difference methods for differential equations numerical analysis of finite volume methods Pure and applied mathematics Differenzenverfahren - Partielle Differentialgleichung Differenzenverfahren (DE-588)4134362-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4134362-1 (DE-588)4044779-0 |
| title | Generalized difference methods for differential equations numerical analysis of finite volume methods |
| title_auth | Generalized difference methods for differential equations numerical analysis of finite volume methods |
| title_exact_search | Generalized difference methods for differential equations numerical analysis of finite volume methods |
| title_full | Generalized difference methods for differential equations numerical analysis of finite volume methods Ronghua Li ; Zhongying Chen ; Wei Wu |
| title_fullStr | Generalized difference methods for differential equations numerical analysis of finite volume methods Ronghua Li ; Zhongying Chen ; Wei Wu |
| title_full_unstemmed | Generalized difference methods for differential equations numerical analysis of finite volume methods Ronghua Li ; Zhongying Chen ; Wei Wu |
| title_short | Generalized difference methods for differential equations |
| title_sort | generalized difference methods for differential equations numerical analysis of finite volume methods |
| title_sub | numerical analysis of finite volume methods |
| topic | Differenzenverfahren - Partielle Differentialgleichung Differenzenverfahren (DE-588)4134362-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differenzenverfahren - Partielle Differentialgleichung Differenzenverfahren Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009153699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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