Difference Methods for Singular Perturbation Problems 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.
The first part of the book explores boundary value problems for elliptic and parabolic reaction-diffusion and convection-diffusion equations in n-dimensional domains with smooth and piecewise-smooth boundaries. The authors develop a technique for constructing and justifying ε uniformly convergent difference schemes for boundary value problems with fewer restrictions on the problem data.
Containing information published mainly in the last four years, the second section focuses on problems with boundary layers and additional singularities generated by nonsmooth data, unboundedness of the domain, and the perturbation vector parameter. This part also studies both the solution and its derivatives with errors that are independent of the perturbation parameters.
Co-authored by the creator of the Shishkin mesh, this book presents a systematic, detailed development of approaches to construct ε uniformly convergent finite difference schemes for broad classes of singularly perturbed boundary value problems.
"This book focuses on the development of robust difference schemes for wide classes of boundary value problems. … The book can be of use for scientists, researchers, students, and 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."
—EMS Newsletter, December 2009
Part I: Grid Approximations of Singular Perturbation Partial Differential Equations
Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Smooth Boundaries
Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Piecewise-Smooth Boundaries
Generalizations for Elliptic Reaction-Diffusion Equations
Parabolic Reaction-Diffusion Equations
Elliptic Convection-Diffusion Equations
Parabolic Convection-Diffusion Equations
Part II: Advanced Trends in ε Uniformly Convergent Difference Methods
Grid Approximations of Parabolic Reaction-Diffusion Equations with Three Perturbation Parameters
Application of Widths for Construction of Difference Schemes for Problems with Moving Boundary Layers
High-Order Accurate Numerical Methods for Singularly Perturbed Problems
A Finite Difference Scheme on a priori Adapted Grids for a Singularly Perturbed Parabolic Convection-Diffusion Equation
On Conditioning of Difference Schemes and Their Matrices for Singularly Perturbed Problems
Approximation of Systems of Singularly Perturbed Elliptic Reaction-Diffusion Equations with Two Parameters