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.

Includes bibliographical references and index.

1. Motivation for the study of singular perturbation problems -- 2. Simple examples of singular perturbation problems -- 3. Numerical methods for singular perturbation problems -- 4. Simple fitted operator methods in one dimension -- 5. Simple fitted mesh methods in one dimension -- 6. Convergence of fitted mesh finite difference methods for linear reaction-diffusion problems in one dimension -- 7. Properties of upwind finite difference operators on piecewise uniform fitted meshes -- 8. Convergence of fitted mesh finite difference methods for linear convection-diffusion problems in one dimension -- 9. Fitted mesh finite element methods for linear convection-diffusion problems in one dimension -- 10. Convergence of Schwarz iterative methods for fitted mesh methods in one dimension -- 11. Linear convection-diffusion problems in two dimensions and their numerical solution -- 12. Bounds on the derivatives of solutions of linear convection-diffusion problems in two dimensions with regular boundary layers -- 13. Convergence of fitted mesh finite difference methods for linear convection-diffusion problems in two dimensions with regular boundary layers -- 14. Limitations of fitted operator methods on uniform rectangular meshes for problems with parabolic boundary layers -- 15. Fitted numerical methods for problems with initial and parabolic boundary layers.

Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods.

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