Numerical approximation of partial differential equations: with 17 tables
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1997
|
| Ausgabe: | 2., corr. printing |
| Schriftenreihe: | Springer series in computational mathematics
23 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 509 - 535 |
| Beschreibung: | XVI, 543 S. Ill., graph. Darst. |
| ISBN: | 3540571116 |
Internformat
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| 245 | 1 | 0 | |a Numerical approximation of partial differential equations |b with 17 tables |c Alfio Quarteroni ; Alberto Valli |
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Datensatz im Suchindex
| _version_ | 1804125777013243904 |
|---|---|
| adam_text | ALFIO QUARTERONI ALBERTO VALLI NUMERICAL APPROXIMATION OF PARTIAL
DIFFERENTIAL EQUATIONS WITH 59 FIGURES AND 17 TABLES SPRINGER 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 12 9 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-GALERKIN 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 II. 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 A-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 STEKLOV-POINCARE 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 AND 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
|
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| id | DE-604.BV011272727 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:06:57Z |
| institution | BVB |
| isbn | 3540571116 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007569900 |
| oclc_num | 845373245 |
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| physical | XVI, 543 S. Ill., graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Quarteroni, Alfio 1952- Verfasser (DE-588)120370158 aut Numerical approximation of partial differential equations with 17 tables Alfio Quarteroni ; Alberto Valli 2., corr. printing Berlin [u.a.] Springer 1997 XVI, 543 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 23 Literaturverz. S. 509 - 535 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Approximation (DE-588)4002498-2 s Numerisches Verfahren (DE-588)4128130-5 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Valli, Alberto Verfasser aut Springer series in computational mathematics 23 (DE-604)BV000012004 23 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007569900&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Quarteroni, Alfio 1952- Valli, Alberto Numerical approximation of partial differential equations with 17 tables Springer series in computational mathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Approximation (DE-588)4002498-2 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4128130-5 (DE-588)4002498-2 |
| title | Numerical approximation of partial differential equations with 17 tables |
| title_auth | Numerical approximation of partial differential equations with 17 tables |
| title_exact_search | Numerical approximation of partial differential equations with 17 tables |
| title_full | Numerical approximation of partial differential equations with 17 tables Alfio Quarteroni ; Alberto Valli |
| title_fullStr | Numerical approximation of partial differential equations with 17 tables Alfio Quarteroni ; Alberto Valli |
| title_full_unstemmed | Numerical approximation of partial differential equations with 17 tables Alfio Quarteroni ; Alberto Valli |
| title_short | Numerical approximation of partial differential equations |
| title_sort | numerical approximation of partial differential equations with 17 tables |
| title_sub | with 17 tables |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Approximation (DE-588)4002498-2 gnd |
| topic_facet | Partielle Differentialgleichung Numerisches Verfahren Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007569900&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
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