Numerical approximation of partial differential equations:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Deutsch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Ausgabe: | 1. softcover print. |
| Schriftenreihe: | Springer series in computational mathematics
23 |
| Schlagwörter: | |
| Links: | http://d-nb.info/989570428/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016990506&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Literaturverz. S. 509 - 535 |
| Umfang: | XVI, 543 S. Ill., graph. Darst. |
| ISBN: | 9783540852674 |
Internformat
MARC
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| 020 | |a 9783540852674 |c Pp. : DM 128.00 |9 978-3-540-85267-4 | ||
| 035 | |a (OCoLC)263409555 | ||
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| 100 | 1 | |a Quarteroni, Alfio |d 1952- |e Verfasser |0 (DE-588)120370158 |4 aut | |
| 245 | 1 | 0 | |a Numerical approximation of partial differential equations |c Alfio Quarteroni ; Alberto Valli |
| 250 | |a 1. softcover print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XVI, 543 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer series in computational mathematics |v 23 | |
| 500 | |a Literaturverz. S. 509 - 535 | ||
| 650 | 0 | 7 | |a Approximation |0 (DE-588)4002498-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Approximation |0 (DE-588)4002498-2 |D s |
| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | 2 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Valli, Alberto |e Verfasser |4 aut | |
| 830 | 0 | |a Springer series in computational mathematics |v 23 |w (DE-604)BV000012004 |9 23 | |
| 856 | 4 | |u http://d-nb.info/989570428/04 |3 Inhaltsverzeichnis | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016990506&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016990506 | |
Datensatz im Suchindex
| _version_ | 1819355242333995008 |
|---|---|
| adam_text | Table
of
Contents
Part
I.
Basic Concepts
and Methods for
PDEs Approximation
1.
Introduction
............................................ 1
1.1
The Conceptual Path Behind the Approximation
......... 2
1.2
Preliminary Notation and Function Spaces
............... 4
1.3
Some Results About Sobolev Spaces
.................... 10
1.4
Comparison Results
................................... 13
2.
Numerical Solution of Linear Systems
................... 17
2.1
Direct Methods
....................................... 17
2.1.1
Banded Systems
................................ 22
2.1.2
Error Analysis
................................. 23
2.2
Generalities on Iterative Methods
....................... 26
2.3
Classical Iterative Methods
............................ 29
2.3.1
Jacobi Method
................................. 29
2.3.2
Gauss-Seidel Method
............................ 31
2.3.3
Relaxation Methods (S.O.R. and S.S.O.R.)
........ 32
2.3.4
Chebyshev Acceleration Method
.................. 34
2.3.5
The Alternating Direction Iterative Method
........ 37
2.4
Modern Iterative Methods
............................. 39
2.4.1
Preconditioned Richardson Method
............... 39
2.4.2
Conjugate Gradient Method
..................... 46
2.5
Preconditioning
...................................... 51
2.6
Conjugate Gradient and Lanczos like Methods for
Non-Symmetric Problems
.............................. 57
2.6.1
GCR, Orthomin and Orthodir Iterations
.......... 57
2.6.2
Arnoldi
and GMRES Iterations
................... 59
2.6.3
Bi-CG,
CGS
and Bi-CGSTAB Iterations
........... 62
2.7
The Multi-Grid Method
............................... 65
2.7.1
The Multi-Grid Cycles
.......................... 65
2.7.2
A Simple Example
.............................. 67
2.7.3
Convergence
................................... 70
2.8
Complements
........................................ 71
XII Table of
Contents
3.
Finite Element
Approximation .......................... 73
3.1 Triangulation ........................................ 73
3.2 Piecewise-Polynomial Subspaces ........................ 74
3.2.1
The Scalar Case
................................ 75
3.2.2
The Vector Case
................................ 76
3.3
Degrees of Freedom and Shape Functions ................
77
3.3.1
The Scalar
Case: Triangular Finite Elements
....... 77
3.3.2
The Scalar Case:
Parallelepipedal
Finite Elements
.. 80
• 3.3.3
The Vector Case
................................ 82
3.4
The Interpolation Operator
............................ 85
3.4.1
Interpolation Error: the Scalar Case
............... 85
3.4.2
Interpolation Error: the Vector Case
.............. 91
3.5
Projection Operators
.................................. 96
3.6
Complements
........................................ 99
4.
Polynomial Approximation
.............................. 101
4.1
Orthogonal Polynomials
............................... 101
4.2
Gaussian Quadrature and Interpolation
.................. 103
4.3
Chebyshev Expansion
................................. 105
4.3.1
Chebyshev Polynomials
......................... 105
4.3.2
Chebyshev Interpolation
......................... 107
4.3.3
Chebyshev Projections
.......................... 113
4.4
Legendre Expansion
................................... 115
4.4.1
Legendre Polynomials
........................... 115
4.4.2
Legendre Interpolation
.......................... 117
4.4.3
Legendre Projections
............................ 120
4.5
Two-Dimensional Extensions
........................... 121
4.5.1
The Chebyshev Case
............................ 121
4.5.2
The Legendre Case
............................. 124
4.6
Complements
........................................ 127
5.
Galerkin, Collocation and Other Methods
............... 129
5.1
An Abstract Reference Boundary Value Problem
......... 129
5.1.1
Some Results of Functional Analysis
.............. 133
5.2
Galerkin Method
..................................... 136
5.3
Petrov-GaJerkin Method
............................... 138
5.4
Collocation Method
................................... 140
5.5
Generalized Galerkin Method
.......................... 141
5.6
Time-Advancing Methods for Time-Dependent Problems
.. 144
5.6.1
Semi-Discrete Approximation
.................... 148
5.6.2
Fully-Discrete Approximation
.................... 148
5.7
Fractional-Step and Operator-Splitting Methods
.......... 151
5.8
Complements
........................................ 156
Table
of Contents
XIII
Part
IL
Approximation of Boundary Value Problems
6.
Elliptic Problems: Approximation by Galerkin and
Collocation Methods
.................................... 159
6.1
Problem Formulation and Mathematical Properties
....... 159
6.1.1
Variational Form of Boundary Value Problems
..... 161
6.1.2
Existence, Uniqueness and
А
-Priori
Estimates
...... 164
6.1.3
Regularity of Solutions
.......................... 167
6.1.4
On the Degeneracy of the Constants in Stability
and Error Estimates
............................ 168
6.2
Numerical Methods: Construction and Analysis
........... 169
6.2.1
Galerkin Method: Finite Element and Spectral
Approximations
................................ 170
6.2.2
Spectral Collocation Method
..................... 179
6.2.3
Generalized Galerkin Method
.................... 187
6.3
Algorithmic Aspects
.................................. 189
6.3.1
Algebraic Formulation
........................... 190
6.3.2
The Finite Element Case
........................ 192
6.3.3
The Spectral Collocation Case
................... 198
6.4
Domain Decomposition Methods
........................ 204
6.4.1
The
Schwarz
Method
............................ 206
6.4.2
Iteration-by-Subdomain Methods Based on
Transmission Conditions at the Interface
.......... 209
6.4.3
The
Stekiov-Poincaré
Operator
___............... 212
6.4.4
The Connection Between Iterations-by-Subdomain
Methods and the
Schur
Complement System
....... 215
7.
Elliptic Problems: Approximation by Mixed and
Hybrid Methods
........................................ 217
7.1
Alternative Mathematical Formulations
.................. 217
7.1.1
The Minimum Complementary Energy Principle
----- 218
7.1.2
Saddle-Point Formulations: Mixed and Hybrid
Methods
....................................... 222
7.2
Approximation by Mixed Methods
...................... 230
7.2.1
Setting up and Analysis
......................... 230
7.2.2
An Example: the Raviart-Thomas Finite Elements
.. 235
7.3
Some Remarks on the Algorithmic Aspects
.............. 241
7.4
The Approximation of More General Constrained
Problems
............................................ 246
7.4.1
Abstract Formulation
........................... 246
7.4.2
Analysis of Stability and Convergence
............. 250
7.4.3
How to Verify the Uniform Compatibility Condition
253
7.5
Complements
........................................ 255
XIV Table of
Contents
8.
Steady Advection-Diffusion Problems
................... 257
8.1
Mathematical Formulation
............................. 257
8.2
A One-Dimensional Example
........................... 258
8.2.1
Galerkin Approximation and Centered Finite
Differences
..................................... 259
8.2.2
Upwind Finite Differences and Numerical Diffusion
. 262
8.2.3
Spectral Approximation
......................... 263
8.3
Stabilization Methods
................................. 265
8.3.1
The Artificial Diffusion Method
.................. 267
8.3.2
Strongly Consistent Stabilization Methods for
Finite Elements
................................ 269
8.3.3
Stabilization by Bubble Functions
................ 273
8.3.4
Stabilization Methods for Spectral Approximation
.. 277
8.4
Analysis of Strongly Consistent Stabilization Methods
..... 280
8.5
Some Numerical Results
............................... 288
8.6
The Heterogeneous Method
............................ 289
9.
The Stokes Problem
.................................... 297
9.1
Mathematical Formulation and Analysis
................. 297
9.2
Galerkin Approximation
............................... 300
9.2.1
Algebraic Form of the Stokes Problem
............. 303
9.2.2
Compatibility Condition and Spurious Pressure
Modes
........................................ 304
9.2.3
Divergence-Free Property and Locking Phenomena
.. 305
9.3
Finite Element Approximation
......................... 306
9.3.1
Discontinuous Pressure Finite Elements
............ 306
9.3.2
Continuous Pressure Finite Elements
.............. 310
9.4
Stabilization Procedures
............................... 311
9.5
Approximation by Spectral Methods
.................... 317
9.5.1
Spectral Galerkin Approximation
................. 319
9.5.2
Spectral Collocation Approximation
............... 323
9.5.3
Spectral Generalized Galerkin Approximation
...... 324
9.6
Solving the Stokes System
............................. 325
9.6.1
The Pressure-Matrix Method
..................... 326
9.6.2
The Uzawa Method
............................. 327
9.6.3
The Arrow-Hurwicz Method
..................... 328
9.6.4
Penalty Methods
............................... 329
9.6.5
The Augmented-Lagrangian Method
.............. 330
9.6.6
Methods Based on Pressure Solvers
............... 331
9.6.7
A Global Preconditioning Technique
.............. 335
9.7
Complements
........................................ 337
10.
The Steady Navier-Stokes Problem
..................... 339
10.1
Mathematical Formulation
............................. 339
Table
of Contents XV
10.1.1
Other Kind of Boundary Conditions
.............. 343
10.1.2
An Abstract Formulation
........................ 345
10.2
Finite Dimensional Approximation
...................... 346
10.2.1
An Abstract Approximate Problem
............... 347
10.2.2
Approximation by Mixed Finite Element Methods
.. 349
10.2.3
Approximation by Spectral Collocation Methods
... 351
10.3
Numerical Algorithms
................................. 353
10.3.1
Newton Methods and the Continuation Method
___ 353
10.3.2
An Operator-Splitting Algorithm
................. 358
10.4
Stream Function-Vorticity Formulation of the
Navier-Stokes Equations
............................... 359
10.5
Complements
........................................ 361
Part III. Approximation of Initial-Boundary Value Problems
11.
Parabolic Problems
..................................... 363
11.1
Initial-Boundary Value Problems
ал
d
Weak Formulation
... 363
11.1.1
Mathematical Analysis of Initial-Boundary Value
Problems
...................................... 365
11.2
Semi-Discrete Approximation
.......................... 373
11.2.1
The Finite Element Case
........................ 373
11.2.2
The Case of Spectral Methods
.................... 379
11.3
Time-Advancing by Finite Differences
................... 384
11.3.1
The Finite Element Case
........................ 385
11.3.2
The Case of Spectral Methods
.................... 396
11.4
Some Remarks on the Algorithmic Aspects
.............. 401
11.5
Complements
........................................ 404
12.
unsteady Advection-Diffusion Problems
................ 405
12.1
Mathematical Formulation
............................. 405
12.2
Time-Advancing by Finite Differences
................... 408
12.2.1
A Sharp Stability Result for the 0-scheme
......... 408
12.2.2
A Semi-Implicit Scheme
......................... 411
12.3
The Discontinuous Galerkin Method for Stabilized
Problems
............................................ 415
12.4
Operator-Splitting Methods
............................ 418
12.5
A Characteristic Galerkin Method
...................... 423
13.
The Unsteady Navier-Stokes Problem
................... 429
13.1
The Navier-Stokes Equations for Compressible and
Incompressible Flows
.................................. 430
13.1.1
Compressible Flows
............................. 431
13.1.2
Incompressible Flows
............................ 432
XVI Table of
Contents
13.2
Mathematical Formulation and Behaviour of Solutions
----- 433
13.3
Semi-Discrete Approximation
.......................... 434
13.4
Time-Advancing by Finite Differences
................... 438
13.5
Operator-Splitting Methods
............................ 441
13.6
Other Approaches
.................................... 446
13.7
Complements
........................................ 448
14.
Hyperbolic Problems
................................... 449
14.1
Some Instances of Hyperbolic Equations
................. 450
14.1.1
Linear Scalar Advection Equations
................ 450
14.1.2
Linear Hyperbolic Systems
....................... 451
14.1.3
Initial-Boundary Value Problems
................. 453
14.1.4
Nonlinear Scalar Equations
...................... 455
14.2
Approximation by Finite Differences
.................... 461
14.2.1
Linear Scalar Advection Equations and Hyperbolic
Systems
....................................... 461
14.2.2
Stability, Consistency, Convergence
............... 465
14.2.3
Nonlinear Scalar Equations
...................... 471
14.2.4
High Order Shock Capturing Schemes
............. 475
14.3
Approximation by Finite Elements
...................... 481
14.3.1
Galerkin Method
............................... 482
14.3.2
Stabilization of the Galerkin Method
.............. 485
14.3.3
Space-Discontinuous Galerkin Method
............. 487
14.3.4
Schemes for Time-Discretization
.................. 488
14.4
Approximation by Spectral Methods
.................... 490
14.4.1
Spectral Collocation Method: the Scalar Case
...... 491
14.4.2
Spectral Collocation Method: the Vector Case
...... 494
14.4.3
Time-Advancing and Smoothing Procedures
....... 496
14.5
Second Order Linear Hyperbolic Problems
............... 497
14.6
The Finite Volume Method
............................ 501
14.7
Complements
........................................ 508
References
.................................................. 509
Subject Index
.............................................. 537
|
| any_adam_object | 1 |
| author | Quarteroni, Alfio 1952- Valli, Alberto |
| author_GND | (DE-588)120370158 |
| author_facet | Quarteroni, Alfio 1952- Valli, Alberto |
| author_role | aut aut |
| author_sort | Quarteroni, Alfio 1952- |
| author_variant | a q aq a v av |
| building | Verbundindex |
| bvnumber | BV035183790 |
| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)263409555 (DE-599)BVBBV035183790 |
| dewey-full | 518.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.64 |
| dewey-search | 518.64 |
| dewey-sort | 3518.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. softcover print. |
| format | Book |
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| id | DE-604.BV035183790 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:25:05Z |
| institution | BVB |
| isbn | 9783540852674 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016990506 |
| oclc_num | 263409555 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-19 DE-BY-UBM DE-83 DE-11 DE-20 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-19 DE-BY-UBM DE-83 DE-11 DE-20 DE-188 |
| physical | XVI, 543 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spellingShingle | Quarteroni, Alfio 1952- Valli, Alberto Numerical approximation of partial differential equations Springer series in computational mathematics Approximation (DE-588)4002498-2 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Numerical approximation of partial differential equations |
| title_auth | Numerical approximation of partial differential equations |
| title_exact_search | Numerical approximation of partial differential equations |
| title_full | Numerical approximation of partial differential equations Alfio Quarteroni ; Alberto Valli |
| title_fullStr | Numerical approximation of partial differential equations Alfio Quarteroni ; Alberto Valli |
| title_full_unstemmed | Numerical approximation of partial differential equations Alfio Quarteroni ; Alberto Valli |
| title_short | Numerical approximation of partial differential equations |
| title_sort | numerical approximation of partial differential equations |
| topic | Approximation (DE-588)4002498-2 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Approximation Partielle Differentialgleichung Numerisches Verfahren |
| url | http://d-nb.info/989570428/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016990506&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
| work_keys_str_mv | AT quarteronialfio numericalapproximationofpartialdifferentialequations AT vallialberto numericalapproximationofpartialdifferentialequations |