Robust Numerical Methods for Singularly Perturbed Differential Equations

by Roos, Hans-Gorg; Stynes, Martin; Tobiska, Lutz

• ISBN13: 9783540344667
• ISBN10: 3540344667
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-11-03
• Publisher: Springer Nature
• More Book Details
• Purchase Benefits

Summary

This considerably extended and completely revised second edition incorporates many new developments in the thriving field of numerical methods for singularly perturbed differential equations. It provides a thorough foundation for the numerical analysis and solution of these problems, which model many physical phenomena whose solutions exhibit layers. The book focuses on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics. It offers a comprehensive overview of suitable numerical methods while emphasizing those with realistic error estimates. The book should be useful for scientists requiring effective numerical methods for singularly perturbed differential equations.

Table of Contents

Notation	p. XIII
Introduction	p. 1
Ordinary Differential Equations
The Analytical Behaviour of Solutions	p. 9
Linear Second-Order Problems Without Turning Points	p. 11
Asymptotic Expansions	p. 12
The Green's Function and Stability Estimates	p. 16
A Priori Estimates for Derivatives and Solution Decomposition	p. 21
Linear Second-Order Turning-Point Problems	p. 25
Quasilinear Problems	p. 29
Linear Higher-Order Problems and Systems	p. 35
Asymptotic Expansions for Higher-Order Problems	p. 35
A Stability Result	p. 36
Systems of Ordinary Differential Equations	p. 38
Numerical Methods for Second-Order Boundary Value Problems	p. 41
Finite Difference Methods on Equidistant Meshes	p. 41
Classical Convergence Theory for Central Differencing	p. 41
Upwind Schemes	p. 45
The Concept of Uniform Convergence	p. 57
Uniformly Convergent Schemes of Higher Order	p. 66
Linear Turning-Point Problems	p. 68
Some Nonlinear Problems	p. 71
Finite Element Methods on Standard Meshes	p. 76
Basic Results for Standard Finite Element Methods	p. 76
Upwind Finite Elements	p. 79
Stabilized Higher-Order Methods	p. 84
Variational Multiscale and Differentiated Residual Methods	p. 95
Uniformly Convergent Finite Element Methods	p. 104
Finite Volume Methods	p. 114
Finite Difference Methods on Layer-adapted Grids	p. 116
Graded Meshes	p. 119
Piecewise Equidistant Meshes	p. 127
Adaptive Strategies Based on Finite Differences	p. 141
Parabolic Initial-Boundary Value Problems in One Space Dimension
Introduction	p. 155
Analytical Behaviour of Solutions	p. 159
Existence, Uniqueness, Comparison Principle	p. 159
Asymptotic Expansions and Bounds on Derivatives	p. 161
Finite Difference Methods	p. 169
First-Order Problems	p. 169
Consistency	p. 169
Stability	p. 171
Convergence in L[subscript 2]	p. 174
Convection-Diffusion Problems	p. 177
Consistency and Stability	p. 178
Convergence	p. 182
Polynomial Schemes	p. 183
Uniformly Convergent Methods	p. 187
Exponential Fitting in Space	p. 188
Layer-Adapted Tensor-Product Meshes	p. 189
Reaction-Diffusion Problems	p. 191
Finite Element Methods	p. 195
Space-Based Methods	p. 196
Polynomial Upwinding	p. 197
Uniformly Convergent Schemes	p. 199
Local Error Estimates	p. 203
Subcharacteristic-Based Methods	p. 205
SDFEM in Space-Time	p. 206
Explicit Galerkin Methods	p. 211
Eulerian-Lagrangian Methods	p. 217
Two Adaptive Methods	p. 223
Streamline Diffusion Methods	p. 223
Moving Mesh Methods (r-refinement)	p. 225
Elliptic and Parabolic Problems in Several Space Dimensions
Analytical Behaviour of Solutions	p. 235
Classical and Weak Solutions	p. 235
The Reduced Problem	p. 238
Asymptotic Expansions and Boundary Layers	p. 243
A Priori Estimates and Solution Decomposition	p. 247
Finite Difference Methods	p. 259
Finite Difference Methods on Standard Meshes	p. 259
Exponential Boundary Layers	p. 259
Parabolic Boundary Layers	p. 266
Layer-Adapted Meshes	p. 268
Exponential Boundary Layers	p. 268
Parabolic Layers	p. 274
Finite Element Methods	p. 277
Inverse-Monotonicity-Preserving Methods Based on Finite Volume Ideas	p. 278
Residual-Based Stabilizations	p. 302
Streamline Diffusion Finite Element Method (SDFEM)	p. 302
Galerkin Least Squares Finite Element Method (GLSFEM)	p. 327
Residual-Free Bubbles	p. 333
Adding Symmetric Stabilizing Terms	p. 338
Local Projection Stabilization	p. 338
Continuous Interior Penalty Stabilization	p. 352
The Discontinuous Galerkin Finite Element Method	p. 363
The Primal Formulation for a Reaction-Diffusion Problem	p. 363
A First-Order Hyperbolic Problem	p. 368
dGFEM Error Analysis for Convection-Diffusion Problems	p. 371
Uniformly Convergent Methods	p. 376
Operator-Fitted Methods	p. 377
Layer-Adapted Meshes	p. 381
Adaptive Methods	p. 407
Adaptive Finite Element Methods for Non-Singularly Perturbed Elliptic Problems: an Introduction	p. 407
Robust and Semi-Robust Residual Type Error Estimators	p. 414
A Variant of the DWR Method for Streamline Diffusion	p. 421
Time-Dependent Problems	p. 427
Analytical Behavior of Solutions	p. 428
Finite Difference Methods	p. 429
Finite Element Methods	p. 434
The Incompressible Navier-Stokes Equations
Existence and Uniqueness Results	p. 449
Upwind Finite Element Method	p. 453
Higher-Order Methods of Streamline Diffusion Type	p. 465
The Oseen Problem	p. 466
The Navier-Stokes Problem	p. 476
Local Projection Stabilization for Equal-Order Interpolation	p. 485
Local Projection Stabilization in an Abstract Setting	p. 486
Convergence Analysis	p. 488
The Special Interpolant	p. 488
Stability	p. 489
Consistency Error	p. 491
A priori Error Estimate	p. 492
Local Projection onto Coarse-Mesh Spaces	p. 498
Simplices	p. 498
Quadrilaterals and Hexahedra	p. 499
Schemes Based on Enrichment of Approximation Spaces	p. 501
Simplices	p. 502
Quadrilaterals and Hexahedra	p. 502
Relationship to Subgrid Modelling	p. 504
Two-Level Approach with Piecewise Linear Elements	p. 505
Enriched Piecewise Linear Elements	p. 507
Spectral Equivalence of the Stabilizing Terms on Simplices	p. 508
Local Projection Method for Inf-Sup Stable Elements	p. 511
Discretization by Inf-Sup Stable Elements	p. 512
Stability and Consistency	p. 514
Convergence	p. 516
Methods of Order r in the Case [sigma] > 0	p. 517
Methods of Order r in the Case [sigma greater than or equal] 0	p. 522
Methods of Order r + 1/2	p. 526
Mass Conservation for Coupled Flow-Transport Problems	p. 529
A Model Problem	p. 529
Continuous and Discrete Mass Conservation	p. 530
Approximated Incompressible Flows	p. 532
Mass-Conservative Methods	p. 534
Higher-Order Flow Approximation	p. 534
Post-Processing of the Discrete Velocity	p. 536
Scott-Vogelius Elements	p. 542
Adaptive Error Control	p. 545
References	p. 551
Index	p. 599
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