The present monograph is devoted to the construction and investigation of new difference schemes for approximating the solutions of regular and singular perturbation boundary-value problems for partial differential equations. This approach permits extending essentially a class of problems where the theory of difference methods is applicable. Namely, it is now possible to investigate differential equations with variable coefficients and regular and singular perturbation boundary-value problems. The investigation is based on new coercivity inequalities.The book will be of value for professional mathematicians as well as for advanced students in the fields of numerical analysis, functional analysis, and differential equations.