In this chapter boundary value problems are considered for elliptic reactiondiffusion equations in domains with smooth boundaries. The construction of finite difference schemes and the justification of their convergence is carried out using an analog of the well-known sufficient condition for convergence of schemes for regular boundary value problems that is a consequence of approximation by a stable finite difference scheme (its description see, e.g., [79, 108, 100]). When applied to singularly perturbed boundary value problems, this sufficient condition for ε-uniform convergence can be stated in the following way:

ε-uniform convergence of a finite difference scheme follows from ε-uniform approximation of the boundary value problem by an ε-uniformly stable finite difference scheme.