1 Introduction ................................................. 1
2 Layer-Adapted Meshes ......................................... 5
2.1 Convection-Diffusion Problems ........................... 6
2.1.1 Bakhvalov Meshes ................................. 7
2.1.2 Shishkin Meshes .................................. 9
2.1.3 Shishkin-Type Meshes ............................ 10
2.1.4 Turning-Point Boundary Layers ................... 15
2.1.5 Interior Layers ................................. 16
2.1.6 Overlapping Layers .............................. 17
2.2 Reaction-Convection-Diffusion Problems ................. 19
2.2.1 Interior Layers ................................. 21
2.2.2 Overlapping Layers .............................. 23
2.3 Two-Dimensional Problems ............................... 24
2.3.1 Reaction-Diffusion Problems ..................... 25
2.3.2 Convection-Diffusion ............................ 26
Part I One Dimensional Problems
3 The Analytical Behaviour of Solutions ....................... 33
3.1 Preliminaries .......................................... 34
3.1.1 Stability of Differential Operators ............. 34
3.1.2 Green's Functions ............................... 36
3.1.3 M-Matrices ...................................... 37
3.2 Reaction-Convection-Diffusion Problems ................. 38
3.2.1 Stability and Green's Function Estimates ........ 39
3.2.2 Derivative Bounds and Solution Decomposition .... 45
3.3 Reaction-Diffusion Problems ............................ 48
3.3.1 Scalar Reaction-Diffusion Problems .............. 48
3.3.2 Systems of Reaction-Diffusion Equations ......... 52
3.4 Convection-Diffusion Problems with Regular Layers ...... 57
3.4.1 Scalar Convection-Diffusion Problems ............ 57
3.4.2 Weakly Coupled Systems .......................... 64
3.4.3 Strongly Coupled Systems ........................ 66
3.5 Convection-Diffusion Problems with Turning-Point
Layers ................................................. 69
3.5.1 Stability ....................................... 69
3.5.2 Derivative Bounds and Solution Decomposition .... 71
4 Finite Difference Schemes for Convection-Diffusion
Problems .................................................... 77
4.1 Notation ............................................... 77
4.2 A Simple Upwind Difference Scheme ...................... 79
4.2.1 Stability of the Discrete Operator .............. 80
4.2.2 A Priori Error Bounds ........................... 84
4.2.3 Error Expansion ................................. 87
4.2.4 A Posteriori Error Estimation and Adaptivity .... 92
4.2.5 An Alternative Convergence Proof ................ 97
4.2.6 The Truncation Error and Barrier Function
Technique ...................................... 100
4.2.7 Discontinuous Coefficients and Point Sources ... 103
4.2.8 Quasilinear Problems ........................... 106
4.2.9 Derivative Approximation ....................... 107
4.3 Second-Order Difference Schemes ....................... 109
4.3.1 Second-Order Upwind Schemes .................... 109
4.3.2 Central Differencing ........................... 119
4.3.3 Convergence Acceleration Techniques ............ 121
4.3.4 A Numerical Example ............................ 132
4.4 Systems ............................................... 134
4.4.1 Weakly Coupled Systems in One Dimension ........ 134
4.4.2 Strongly Coupled Systems ....................... 137
4.5 Problems with Turning Point Layers .................... 143
4.5.1 A First-Order Upwind Scheme .................... 144
4.5.2 Convergence on Shishkin Meshes ................. 147
4.5.3 A Numerical Example ............................ 148
5 Finite Element and Finite Volume Methods ................... 151
5.1 The Interpolation Error ............................... 152
5.2 Linear Galerkin FEM ................................... 154
5.2.1 Convergence .................................... 154
5.2.2 Supercloseness ................................. 156
5.2.3 Gradient Recovery and a Posteriori Error
Estimation ..................................... 160
5.2.4 A Numerical Example ............................ 162
5.3 Stabilised FEM ........................................ 163
5.3.1 Artificial Viscosity Stabilisation ............. 163
5.3.2 Streamline-Diffusion Stabilisation ............. 164
5.4 An Upwind Finite Volume Method ....................... 168
5.4.1 Stability of the FVM ........................... 171
5.4.2 Convergence in the Energy Norm ................. 175
5.4.3 Convergence in the Maximum Norm ................ 180
5.4.4 A Numerical Example ............................ 182
6 Discretisations of Reaction-Convection-Diffusion
Problems ................................................... 183
6.1 Reaction-Diffusion .................................... 183
6.1.1 Linear Finite Elements ......................... 184
6.1.2 Central Differencing ........................... 190
6.1.3 A Non-Monotone Scheme .......................... 202
6.1.4 A Compact Fourth-Order Scheme .................. 206
6.2 Systems of Reaction-Diffusion Type .................... 214
6.2.1 The Interpolation Error ........................ 214
6.2.2 Linear Finite Elements ......................... 215
6.2.3 Central Differencing ........................... 217
6.3 Reaction-Convection-Diffusion ......................... 221
6.3.1 The Interpolation Error ........................ 222
6.3.2 Simple Upwinding ............................... 223
Part II Two Dimensional Problems
7 The Analytical Behaviour of Solutions ...................... 235
7.1 Preliminaries ......................................... 235
7.1.1 Stability ...................................... 236
7.1.2 Regularity of Solutions ........................ 237
7.2 Reaction-Diffusion .................................... 238
7.2.1 Stability ...................................... 239
7.2.2 Derivative Bounds .............................. 240
7.3 Convection-Diffusion .................................. 243
7.3.1 Regular Layers ................................. 243
7.3.2 Characteristic Layers .......................... 245
8 Reaction-Diffusion Problems ................................ 247
8.1 Central Differencing .................................. 247
8.1.1 Stability ...................................... 248
8.1.2 Convergence on Layer-Adapted Meshes ............ 249
8.1.3 Numerical Results .............................. 253
8.2 Arbitrary Bounded Domains ............................. 254
9 Convection-Diffusion Problems .............................. 257
9.1 Upwind Difference Schemes ............................ 257
9.1.1 Stability ...................................... 258
9.1.2 Pointwise Error Bounds ......................... 258
9.1.3 Error Expansion ................................ 262
9.2 Finite Element Methods ................................ 263
9.2.1 The Interpolation Error ........................ 264
9.2.2 Galerkin FEM ................................... 267
9.2.3 Artificial Viscosity Stabilisation ............. 285
9.2.4 Streamline-Diffusion FEM ....................... 289
9.2.5 Characteristic Layers .......................... 294
9.3 Finite Volume Methods ................................. 297
9.3.1 Coercivity of the Method ....................... 299
9.3.2 Inverse Monotonicity ........................... 302
9.3.3 Convergence .................................... 306
Conclusions and Outlook ....................................... 309
References .................................................... 311
Index ......................................................... 319
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