Grid approximation of a singularly perturbed elliptic convection–diffusion equation in an unbounded domain

Abstract

We consider the Dirichlet problem for a singularly perturbed elliptic convection-diffusion equation in the quarter plane {(x1, x2) : x1, x2 ≥ 0}. The highest derivatives of the equation and the first derivative along the x2-axis contain, respectively, the parameters ε1 and ε2, which take arbitrary values from the half-open interval (0,1] and the segment [–1,1]. For small values of the parameter ε1, a boundary layer appears in the neighbourhood of the domain boundary. Depending on the ratio between the parameters ε1 and ε2, this layer may be regular, elliptic, parabolic, or hyperbolic. Beside the boundary layer scale controlled by the perturbation parameters, one can observe a resolution scale, which is specified by the 'width' of the domain in which the problem is to be solved on a computer. It turns out that for solutions of the boundary value problem and a formal difference scheme (i.e., a scheme on meshes with an infinite number of nodes) considered on bounded subdomains of interest (referred to as resolution subdomains), domains of essential dependence, i.e., domains outside which the finite variation of the solution causes relatively small perturbations of the solution on resolution subdomains, are bounded uniformly with respect to the vector parameter = (ε1, ε2). Using the concept of a 'domain of essential solution dependence', we develop a constructive finite difference scheme (i.e., a scheme on meshes with a finite number of nodes) that converges -uniformly on the bounded resolution subdomains.