Difference Methods for Singular Perturbation Problems
This book focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the ε-uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods. Part I (Grid approximations of singular perturbation partial differential equations) begins with an introduction containing a short history of the field and its main ideas, the principles and the main problems encountered in the construction of the special schemes in the book. In further chapters of Part 1, the following problems are considered: BVP for elliptic reaction-diffusion equations in domains with smooth and piecewise-smooth boundaries, some generalisations, parabolic reaction-diffusion equations, elliptic convection-diffusion equations and parabolic convection-diffusion equations. Part II (Advanced trends in ε-uniformly convergent difference methods) contains material mainly published in the last four years. This includes problems with boundary layers and additional singularities generated by nonsmooth data, unboundedness of the domain and also by the presence of the perturbation vector parameter. Another aspect considered in this part is that both the solution and its derivatives are found with errors that are independent of the perturbation parameters. The book can be of use for scientists and researchers, both for students and for professionals in the field of developing numerical methods for singularly perturbed problems and also for anybody interested in mathematical modelling or in the fields where the problems with boundary and interior layers arise naturally.