Grid approximation for singularly perturbed boundary-value problems with sharply changing density of sources

The Dirichlet problem for an elliptic equation is treated on the n -dimensional layer composed of several sublayers. The terms with the higher-order derivatives in the equation contain (as a factor) a parameter taking arbitrary values in the ray (Ο,οο). The parameter may take different values on different layers that comprise the domain, i.e. it may be discontinuous. At the zero value of the parameter the differential equation degenerates into the zero-order one. The right-hand side of the differential equation (the density of sources) undergoes sharp changes (remaining smooth in each sublayer). Internal intermediate layers appear near the boundaries of subdomains as well as in the vicinity of the points where the density of sources changes sharply. It has been shown that even in the one-dimensional case no fitting method schemes converging uniformly in the parameter exist for such boundary-value problems, if one uses the data of a boundary-value problem (the coefficients and the free term of the equation) taken at a finite number of grid nodes. Using the method of specially condensing grids, the difference schemes converging uniformly in the parameter have been constructed. Many problems in mathematical modelling give rise to the boundary-value problems for singularly perturbed equations treated on compound domains. The higher-order derivatives in the equations contain a parameter (a perturbation) that may be as small as desired. Such problems arise, for example, in modelling and analysing massand heat-transfer processes in laminated media where the massand heat-transfer coefficients strongly differ from layer to layer [7,19]. Localized mass and heat sources distributed over relatively narrow regions may occur in these layers with different physical properties. For small parameters internal intermediate layers appear near the boundaries of subdomains and boundary layers are produced near the boundary of the domain. Besides, internal intermediate layers also appear in the vicinity of lumped and sharply varying sources. This paper deals with the Dirichlet problem for a singularly perturbed elliptic equation treated on a compound domain. In contrast to the boundary-value problems on compound domains [14,17,18] in the present case the values of the parameter in different subdomains are, generally speaking, different and not related to each other. The parameter may vary along the ray (Ο,οο). The density of sources (the right-hand side of the differential equation) sharply changes (remaining smooth in a subdomain) within narrow regions. The parameter may tend to zero in one subdomain and to infinity in another. In such a problem the boundary and intermediate layers appear only in the subdomains where the parameter tends to zero. Therefore, the solution to the boundary-value problem under consideration possesses a restricted smoothness that further degrades as the parameter approaches zero. The accuracy of the classic difference methods depends on the smoothness of the sought-for solution [8,9,11,12]. The accuracy of an approximate solution worsens as its smoothness decreases. In the case of singularly perturbed boundary-value problems the accuracy is lost completely at small values of the parameter [1,2,4], Therefore,