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Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems 2nd ed. 2008 [Kõva köide]

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  • Formaat: Hardback, 604 pages, kõrgus x laius: 235x155 mm, kaal: 1088 g, 41 Illustrations, black and white; XIV, 604 p. 41 illus., 1 Hardback
  • Sari: Springer Series in Computational Mathematics 24
  • Ilmumisaeg: 09-Sep-2008
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540344667
  • ISBN-13: 9783540344667
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Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems 2nd ed. 2008Suurem pilt
  • Formaat: Hardback, 604 pages, kõrgus x laius: 235x155 mm, kaal: 1088 g, 41 Illustrations, black and white; XIV, 604 p. 41 illus., 1 Hardback
  • Sari: Springer Series in Computational Mathematics 24
  • Ilmumisaeg: 09-Sep-2008
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540344667
  • ISBN-13: 9783540344667
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783540344667.html
Märksõnad:
The analysis of singular perturbed di erential equations began early in the twentieth century, when approximate solutions were constructed from asy- totic expansions. (Preliminary attempts appear in the nineteenth century - see[ vD94].)Thistechniquehas ourishedsincethemid-1960sanditsprincipal ideas and methods are described in several textbooks; nevertheless, asy- totic expansions may be impossible to construct or may fail to simplify the given problem and then numerical approximations are often the only option. Thesystematicstudyofnumericalmethodsforsingularperturbationpr- lems started somewhat later - in the 1970s. From this time onwards the - search frontier has steadily expanded, but the exposition of new developments in the analysis of these numerical methods has not received its due attention. The ?rst textbook that concentrated on this analysis was [ DMS80], which collected various results for ordinary di erential equations. But after 1980 no further textbook appeared until 1996, when three books were published: Miller et al. [ MOS96], which specializes in upwind ?nite di erence methods on Shishkin meshes, Morton's book [ Mor96], which is a general introduction to numerical methods for convection-di? usion problems with an emphasis on the cell-vertex ?nite volume method, and [ RST96], the ?rst edition of the present book. Nevertheless many methods and techniques that are important today, especially for partial di erential equations, were developed after 1996.

Arvustused

From the reviews of the second edition:









"It is based on the classical technique of constructing asymptotic solutions to singular perturbation problems . Singular perturbations occur in many important applications and developing asymptotic methods to study such problems continues to present challenges to the applied mathematician. The authors are to be commended for this important book, which describes a very fruitful and extensive interconnected international activity." (Robert E. OMalley, SIAM Reviews, Vol. 51 (2), June, 2009)



"This book gives a survey of recent work on the numerical solution of singular-perturbation problems, mostly for convection-diffusion equations but also for reaction-diffusion equations. One valuable feature of the book is the large number of remarks, which clarify details of the various methods. The book is an essential reference for the researcher on computation of singular perturbation problems." (Gerald W. Hedstrom, Zentralblatt MATH, Vol. 1155, 2009)



"This book collects together some recent results in the area of numerical methods for singularly perturbed differential equations. This well-written and lucid book will act as a useful state-of-the-art reference guide for researchers and students interested in understanding what has been published on robust numerical methods for singularly perturbed differential equations. In addition, it is clear from this book that many avenues of research remain open within the broad field of singularly perturbed problems." (Eugene O'Riordan, Mathematical Reviews, Issue 2009 f)

Notation XIII
Introduction 1
Part I Ordinary Differential Equations
1 The Analytical Behaviour of Solutions
9
1.1 Linear Second-Order Problems Without Turning Points
11
1.1.1 Asymptotic Expansions
12
1.1.2 The Green's Function and Stability Estimates
16
1.1.3 A Priori Estimates for Derivatives and Solution Decomposition
21
1.2 Linear Second-Order Turning-Point Problems
25
1.3 Quasilinear Problems
29
1.4 Linear Higher-Order Problems and Systems
35
1.4.1 Asymptotic Expansions for Higher-Order Problems
35
1.4.2 A Stability Result
36
1.4.3 Systems of Ordinary Differential Equations
38
2 Numerical Methods for Second-Order Boundary Value Problems
41
2.1 Finite Difference Methods on Equidistant Meshes
41
2.1.1 Classical Convergence Theory for Central Differencing
41
2.1.2 Upwind Schemes
45
2.1.3 The Concept of Uniform Convergence
57
2.1.4 Uniformly Convergent Schemes of Higher Order
66
2.1.5 Linear Turning-Point Problems
68
2.1.6 Some Nonlinear Problems
71
2.2 Finite Element Methods on Standard Meshes
76
2.2.1 Basic Results for Standard Finite Element Methods
76
2.2.2 Upwind Finite Elements
79
2.2.3 Stabilized Higher-Order Methods
84
2.2.4 Variational Multiscale and Differentiated Residual Methods
95
2.2.5 Uniformly Convergent Finite Element Methods
104
2.3 Finite Volume Methods
114
2.4 Finite Difference Methods on Layer-adapted Grids
116
2.4.1 Graded Meshes
119
2.4.2 Piecewise Equidistant Meshes
127
2.5 Adaptive Strategies Based on Finite Differences
141
Part II Parabolic Initial-Boundary Value Problems in One Space Dimension
1 Introduction
155
2 Analytical Behaviour of Solutions
159
2.1 Existence, Uniqueness, Comparison Principle
159
2.2 Asymptotic Expansions and Bounds on Derivatives
161
3 Finite Difference Methods
169
3.1 First-Order Problems
169
3.1.1 Consistency
169
3.1.2 Stability
171
3.1.3 Convergence in L2
174
3.2 Convection-Diffusion Problems
177
3.2.1 Consistency and Stability
178
3.2.2 Convergence
182
3.3 Polynomial Schemes
183
3.4 Uniformly Convergent Methods
187
3.4.1 Exponential Fitting in Space
188
3.4.2 Layer-Adapted Tensor-Product Meshes
189
3.4.3 Reaction-Diffusion Problems
191
4 Finite Element Methods
195
4.1 Space-Based Methods
196
4.1.1 Polynomial Upwinding
197
4.1.2 Uniformly Convergent Schemes
199
4.1.3 Local Error Estimates
203
4.2 Subcharacteristic-Based Methods
205
4.2.1 SDFEM in Space-Time
206
4.2.2 Explicit Galerkin Methods
211
4.2.3 Eulerian-Lagrangian Methods
217
5 Two Adaptive Methods
223
5.1 Streamline Diffusion Methods
223
5.2 Moving Mesh Methods (r-refinement)
225
Part III Elliptic and Parabolic Problems in Several Space Dimensions
1 Analytical Behaviour of Solutions
235
1.1 Classical and Weak Solutions
235
1.2 The Reduced Problem
238
1.3 Asymptotic Expansions and Boundary Layers
243
1.4 A Priori Estimates and Solution Decomposition
247
2 Finite Difference Methods
259
2.1 Finite Difference Methods on Standard Meshes
259
2.1.1 Exponential Boundary Layers
259
2.1.2 Parabolic Boundary Layers
266
2.2 Layer-Adapted Meshes
268
2.2.1 Exponential Boundary Layers
268
2.2.2 Parabolic Layers
274
3 Finite Element Methods
277
3.1 Inverse-Monotonicity-Preserving Methods Based on Finite Volume Ideas
278
3.2 Residual-Based Stabilizations
302
3.2.1 Streamline Diffusion Finite Element Method (SDFEM)
302
3.2.2 Galerkin Least Squares Finite Element Method (GLSFEM)
327
3.2.3 Residual-Free Bubbles
333
3.3 Adding Symmetric Stabilizing Terms
338
3.3.1 Local Projection Stabilization
338
3.3.2 Continuous Interior Penalty Stabilization
352
3.4 The Discontinuous Galerkin Finite Element Method
363
3.4.1 The Primal Formulation for a Reaction-Diffusion Problem
363
3.4.2 A First-Order Hyperbolic Problem
368
3.4.3 dGFEM Error Analysis for Convection-Diffusion Problems
371
3.5 Uniformly Convergent Methods
376
3.5.1 Operator-Fitted Methods
377
3.5.2 Layer-Adapted Meshes
381
3.6 Adaptive Methods
407
3.6.1 Adaptive Finite Element Methods for Non-Singularly Perturbed Elliptic Problems: an Introduction
407
3.6.2 Robust and Semi-Robust Residual Type Error Estimators
414
3.6.3 A Variant of the DWR Method for Streamline Diffusion
421
4 Time-Dependent Problems
427
4.1 Analytical Behaviour of Solutions
428
4.2 Finite Difference Methods
429
4.3 Finite Element Methods
434
Part IV The Incompressible Navier-Stokes Equations
1 Existence and Uniqueness Results
449
2 Upwind Finite Element Method
453
3 Higher-Order Methods of Streamline Diffusion Type
465
3.1 The Oseen Problem
466
3.2 The Navier-Stokes Problem
476
4 Local Projection Stabilization for Equal-Order Interpolation
485
4.1 Local Projection Stabilization in an Abstract Setting
486
4.2 Convergence Analysis
488
4.2.1 The Special Interpolant
488
4.2.2 Stability
489
4.2.3 Consistency Error
491
4.2.4 A priori Error Estimate
492
4.3 Local Projection onto Coarse-Mesh Spaces
498
4.3.1 Simplices
498
4.3.2 Quadrilaterals and Hexahedra
499
4.4 Schemes Based on Enrichment of Approximation Spaces
501
4.4.1 Simplices
502
4.4.2 Quadrilaterals and Hexahedra
502
4.5 Relationship to Subgrid Modelling
504
4.5.1 Two-Level Approach with Piecewise Linear Elements
505
4.5.2 Enriched Piecewise Linear Elements
507
4.5.3 Spectral Equivalence of the Stabilizing Terms on Simplices
508
5 Local Projection Method for Inf-Sup Stable Elements
511
5.1 Discretization by Inf-Sup Stable Elements
512
5.2 Stability and Consistency
514
5.3 Convergence
516
5.3.1 Methods of Order r in the Case σ greater than 0
517
5.3.2 Methods of Order r in the Case σ greater than or = to 0
522
5.3.3 Methods of Order r + 1/2
526
6 Mass Conservation for Coupled Flow-Transport Problems
529
6.1 A Model Problem
529
6.2 Continuous and Discrete Mass Conservation
530
6.3 Approximated Incompressible Flows
532
6.4 Mass-Conservative Methods
534
6.4.1 Higher-Order Flow Approximation
534
6.4.2 Post-Processing of the Discrete Velocity
536
6.4.3 Scott-Vogelius Elements
542
7 Adaptive Error Control
545
References 551
Index 599