Hans-Görg Roos
Martin Stynes
Lutz Tobiska
Robust Numerical Methods for
Singularly Perturbed Differential
Equations
SPIN Springer’s internal project number, if known
Convection-Diffusion-Reaction and Flow
Problems
May 27, 2008
Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan Paris Tokyo
Preface
The analysis of singular perturbed differential equations began early in the
twentieth century, when approximate solutions were constructed from asymptotic expansions. (Preliminary attempts appear in the nineteenth century –
see [vD94].) This technique has flourished since the mid-1960s and its principal
ideas and methods are described in several textbooks; nevertheless, asymptotic expansions may be impossible to construct or may fail to simplify the
given problem and then numerical approximations are often the only option.
The systematic study of numerical methods for singular perturbation problems started somewhat later – in the 1970s. From this time onwards the research frontier has steadily expanded, but the exposition of new developments
in the analysis of these numerical methods has not received its due attention.
The first textbook that concentrated on this analysis was [DMS80], which
collected various results for ordinary differential equations. But after 1980
no further textbook appeared until 1996, when three books were published:
Miller et al. [MOS96], which specializes in upwind finite difference methods
on Shishkin meshes, Morton’s book [Mor96], which is a general introduction
to numerical methods for convection-diffusion problems with an emphasis on
the cell-vertex finite volume method, and [RST96], the first edition of the
present book. Nevertheless many methods and techniques that are important
today, especially for partial differential equations, were developed after 1996.
To give some examples, layer-adapted special meshes are frequently used,
new stabilization techniques (such as discontinuous Galerkin methods and local subspace projections) are prominent, and there is a growing interest in
the use of adaptive methods. Consequently contemporary researchers must
comb the literature to gain an overview of current developments in this active
area. In this second edition we retain the exposition of basic material that
underpinned the first edition while extending its coverage to significant new
numerical methods for singularly perturbed differential equations.
Our purposes in writing this introductory book are twofold. First, we
present a structured and comprehensive account of current ideas in the numerical analysis of singularly perturbed differential equations. Second, this
VI
Preface
important area has many open problems and we hope that our book will
stimulate their investigation. Our choice of topics is inevitably personal and
reflects our own main interests.
We have learned a great deal about singularly perturbed problems from
other researchers. We thank those colleagues who helped and influenced
us; these include V.B. Andreev, A.E. Berger, P.A. Farrell, A. Felgenhauer,
E.C. Gartland, Ch. Großmann, A.F. Hegarty, R.B. Kellogg, N. Kopteva,
G. Lube, N. Madden, G. Matthies, J.J.H. Miller, K.W. Morton, F. Schieweck,
G.I. Shishkin, E. Süli, and R. Vulanović; in particular Herbert Goering and
Eugene O’Riordan guided our initial steps in the area. Our research colleague
T. Linß deserves additional thanks for providing many of the figures in this
book.
Our work was supported by the Deutsche Forschungsgemeinschaft and by
the Boole Centre for Research in Informatics at the National University of
Ireland, Cork. We are grateful to them, to the Mathematisches Forschungsinstitut in Oberwolfach for its hospitality, and to Springer-Verlag for its cooperation.
Contents
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Part I Ordinary Differential Equations
1
2
The Analytical Behaviour of Solutions . . . . . . . . . . . . . . . . . . . . .
1.1 Linear Second-Order Problems Without Turning Points . . . . . .
1.1.1 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 The Green’s Function and Stability Estimates . . . . . . . . .
1.1.3 A Priori Estimates for Derivatives and Solution
Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Linear Second-Order Turning-Point Problems . . . . . . . . . . . . . . .
1.3 Quasilinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Linear Higher-Order Problems and Systems . . . . . . . . . . . . . . . . .
1.4.1 Asymptotic Expansions for Higher-Order Problems . . . .
1.4.2 A Stability Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Systems of Ordinary Differential Equations . . . . . . . . . . .
9
11
12
16
21
25
29
35
35
36
38
Numerical Methods for Second-Order Boundary Value
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Finite Difference Methods on Equidistant Meshes . . . . . . . . . . . .
2.1.1 Classical Convergence Theory for Central Differencing .
2.1.2 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 The Concept of Uniform Convergence . . . . . . . . . . . . . . . .
2.1.4 Uniformly Convergent Schemes of Higher Order . . . . . . .
2.1.5 Linear Turning-Point Problems . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Some Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Finite Element Methods on Standard Meshes . . . . . . . . . . . . . . .
2.2.1 Basic Results for Standard Finite Element Methods . . . .
41
41
41
45
57
66
68
71
76
76
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Contents
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
Contents
5
IX
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
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Contents
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 Interpolation485
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 > 0 . . . . . . . . . . . . . . . 517
5.3.2 Methods of Order r in the Case σ ≥ 0 . . . . . . . . . . . . . . . 522
5.3.3 Methods of Order r + 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . 526
Notation
XI
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