Read more
This considerably extended and completely revised second edition incorporates many new developments in the thriving field of numerical methods for singularly perturbed differential equations. It provides a thorough foundation for the numerical analysis and solution of these problems, which model many physical phenomena whose solutions exhibit layers. The book focuses on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics. It offers a comprehensive overview of suitable numerical methods while emphasizing those with realistic error estimates. The book should be useful for scientists requiring effective numerical methods for singularly perturbed differential equations.
List of contents
Ordinary Differential Equations.- The Analytical Behaviour of Solutions.- Numerical Methods for Second-Order Boundary Value Problems.- Parabolic Initial-Boundary Value Problems in One Space Dimension.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Two Adaptive Methods.- Elliptic and Parabolic Problems in Several Space Dimensions.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Time-Dependent Problems.- The Incompressible Navier-Stokes Equations.- Existence and Uniqueness Results.- Upwind Finite Element Method.- Higher-Order Methods of Streamline Diffusion Type.- Local Projection Stabilization for Equal-Order Interpolation.- Local Projection Method for Inf-Sup Stable Elements.- Mass Conservation for Coupled Flow-Transport Problems.- Adaptive Error Control.
Summary
Beginning with ordinary differential equations, then moving on to parabolic and elliptic problems and culminating with the Navier-Stokes equations, the reader is led through the theoretical and practical aspects of the most important methods used to compute numerical solutions for singular perturbation problems.
Additional text
From the reviews of the second edition:
"It is based on the classical technique of constructing asymptotic solutions to singular perturbation problems … . Singular perturbations occur in many important applications and developing asymptotic methods to study such problems continues to present challenges to the applied mathematician. … The authors are to be commended for … this important book, which describes a very fruitful and extensive interconnected international activity." (Robert E. O’Malley, SIAM Reviews, Vol. 51 (2), June, 2009)
"This book gives a survey of recent work on the numerical solution of singular-perturbation problems, mostly for convection-diffusion equations but also for reaction-diffusion equations. … One valuable feature of the book is the large number of remarks, which clarify details of the various methods. … The book is an essential reference for the researcher on computation of singular perturbation problems." (Gerald W. Hedstrom, Zentralblatt MATH, Vol. 1155, 2009)
"This book collects together some recent results in the area of numerical methods for singularly perturbed differential equations. … This well-written and lucid book will act as a useful state-of-the-art reference guide for researchers and students interested in understanding what has been published on robust numerical methods for singularly perturbed differential equations. In addition, it is clear from this book that many avenues of research remain open within the broad field of singularly perturbed problems." (Eugene O'Riordan, Mathematical Reviews, Issue 2009 f)
Report
From the reviews of the second edition:
"It is based on the classical technique of constructing asymptotic solutions to singular perturbation problems ... . Singular perturbations occur in many important applications and developing asymptotic methods to study such problems continues to present challenges to the applied mathematician. ... The authors are to be commended for ... this important book, which describes a very fruitful and extensive interconnected international activity." (Robert E. O'Malley, SIAM Reviews, Vol. 51 (2), June, 2009)
"This book gives a survey of recent work on the numerical solution of singular-perturbation problems, mostly for convection-diffusion equations but also for reaction-diffusion equations. ... One valuable feature of the book is the large number of remarks, which clarify details of the various methods. ... The book is an essential reference for the researcher on computation of singular perturbation problems." (Gerald W. Hedstrom, Zentralblatt MATH, Vol. 1155, 2009)
"This book collects together some recent results in the area of numerical methods for singularly perturbed differential equations. ... This well-written and lucid book will act as a useful state-of-the-art reference guide for researchers and students interested in understanding what has been published on robust numerical methods for singularly perturbed differential equations. In addition, it is clear from this book that many avenues of research remain open within the broad field of singularly perturbed problems." (Eugene O'Riordan, Mathematical Reviews, Issue 2009 f)
