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This book collects, explains and analyses basic methods and recent results for the successful numerical solution of singularly perturbed differential equations. Such equations model many physical phenomena and their solutions are characterized by the presence of layers. The book is a wide-ranging introduction to the exciting current literature in this area. It concentrates on linear convection-diffusion equations and related nonlinear flow problems, encompassing both ordinary and partial differential equations. While many numerical methods are considered, particular attention is paid to those with realistic error estimates. The book provides a solid and thorough foundation for the numerical analysis and solution of singular perturbation problems.
List of contents
I. Ordinary Differential Equations: The analytical behaviour of solutions - numerical methods for second-order boundary value problems - numerical methods for higher-order problems
II. Parabolic Initial-Boundary Value Problems in One Space Dimension: Analytical behaviour of solutions - finite difference methods - finite element methods - adaptive methods
III. Elliptic Boundary Value Problems: Analytical behaviour of solutions - finite difference methods - finite element methods
IV. Incompressible Navier-Stokes Equations: Existence and uniqueness results - an upwind finite element method - stabilized higher order methods - adaptive error control
Appendix: Robust Solvers for Linear Systems
