 | Miller J.J.H. Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions / J.J.H.Miller, E.O'Riordan, G.I.Shishkin. - Singapore; Hackensack: World Scientific, 2012. - xiv, 176 p.: ill. - Bibliogr.: p.169-173. - Ind.: p.175-176. - ISBN-10 981-4390-73-9; ISBN-13 978-981-4390-73-6
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Preface ....................................................... vii
Notation, Terminology and Acknowledgments ...................... xi
1 Motivation for the Study of Singular Perturbation Problems ... 1
2 Simple Examples of Singular Perturbation Problems ............ 5
3 Numerical Methods for Singular Perturbation Problems ........ 13
4 Simple Fitted Operator Methods in One Dimension ............. 21
5 Simple Fitted Mesh Methods in One Dimension ................. 35
6 Convergence of Fitted Mesh Finite Difference Methods
for Linear Reaction-Diffusion Problems in One Dimension ..... 45
7 Properties of Upwind Finite Difference Operators on
Piecewise Uniform Fitted Meshes ............................. 55
8 Convergence of Fitted Mesh Finite Difference Methods for
Linear Convection-Diffusion Problems in One Dimension ....... 63
9 Fitted Mesh Finite Element Methods for Linear
Convection-Diffusion Problems in One Dimension .............. 77
10 Convergence of Schwarz Iterative Methods for Fitted
Mesh Methods in One Dimension ............................... 91
11 Linear Convection-Diffusion Problems in Two Dimensions and
Their Numerical Solution ................................... 105
12 Bounds on the Derivatives of Solutions of Linear
Convection-Diffusion Problems in Two Dimensions with
Regular Boundary Layers .................................... 117
13 Convergence of Fitted Mesh Finite Difference Methods
for Linear Convection-Diffusion Problems in Two
Dimensions with Regular Boundary Layers .................... 127
14 Limitations of Fitted Operator Methods on Uniform
Rectangular Meshes for Problems with Parabolic Boundary
Layers ..................................................... 133
15 Fitted Numerical Methods for Problems with Initial and
Parabolic Boundary Layers .................................. 151
Appendix A Some a priori Bounds for Differential Equations
in Two Dimensions .......................................... 163
Bibliography .................................................. 169
Index ......................................................... 175
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