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 {(x 1 , x 2 ) : x 1 , x 2 0}. The highest derivatives of the equation and the first derivative along the x 2 -axis contain, respectively, the parameters ε 1 and ε 2 , which take arbitrary values from the half-open interval (0, 1] and the segment [−1, 1]. For small values of the parameter ε 1 , a boundary layer appears in the neighbourhood of the domain boundary. Depending on the ratio between the parameters ε 1 and ε 2 , 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 ε = (ε 1 , ε 2 ). 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 ε-uniformly on the bounded resolution subdomains. ) (M ( j.k) , G h( j.k) ) means that this operator (constant, grid) is introduced in formula ( j.k). † Hereafter, M, M i (or m) denote sufficiently large (small) positive constants independent of the vector parameter ε and the parameters of difference schemes.

Superconvergence analysis of a finite element method for a two-parameter singularly perturbed problem

“…The fundamental character of the solutions of the problem (2.1) is significantly different to the character of the solutions of problems considered in [16,17,[23][24][25]. Shishkin considers elliptic problem posed on bounded [25] and unbounded [23,24] domain for which the totally reduced equation (when ε 1 = ε 2 = 0) is of the first order and the presence of various kinds of boundary layers (exponential, parabolic and hyperbolic) is possible.…”

“…Shishkin considers elliptic problem posed on bounded [25] and unbounded [23,24] domain for which the totally reduced equation (when ε 1 = ε 2 = 0) is of the first order and the presence of various kinds of boundary layers (exponential, parabolic and hyperbolic) is possible. Two-parameter elliptic equation whose solution exhibits only exponential layers is considered in O'Riordan at al.…”

“…In contrast to [16,17,[23][24][25], where finite differences are used to obtain a numerical solution, to the best of our knowledge [29] is the only paper considering a finite element method for a two-parameter singularly perturbed problem in two dimensions.…”

An elliptic singularly perturbed problem with two parameters II: Robust finite element solution

“…In the same manner we estimate other layer components. On 0 , we have corresponding mesh sizes (23), (24) and √ meas( 0 ) = √ 0 . Using (32) and S 2 C we estimate the regular term S :…”

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

“…The fundamental character of the solutions of such singularly perturbed problems are significantly different to the character of the solutions of problems from the class (1). Shishkin has examined problems involving differential equations of the form (1a) on bounded [19] and unbounded [20] domains, where the convective term is of the form (a 1 (x, y), μa 2 (x, y)) · ∇u. Note that the fully reduced problem ((ε, μ) = (0, 0)) in [19] is different to the fully reduced problem in the present paper.…”