Grid approximation of a singularly perturbed one-dimensional heat equation on an unbounded domain excluding a rectangle with sides noncollinear to the axes

Abstract

We consider discrete approximations of the first initial boundary value problem for a singularly perturbed heat equation on a domain outside the 'inclusion', i.e. on the domain = R ×[ t 0 , T ] excluding a rectangle whose sides are noncollinear to the x-, t -axes. The solution of the problem has boundary moving layers, regular by their behaviour, in a neighbourhood of the outside part of the boundary and parabolic transition layers in a neighbourhood of characteristics tangent to the inclusion. In addition, the solution of the problem has a discontinuity-type singularity occurring at the instant the inclusion appears. Using the fitted operator method (in a neighbourhood of the layer appearance, where the singularity develops) and the refining mesh technique (in a neighbourhood of the layers), we construct a special difference scheme convergent ε-uniformly almost everywhere in the grid domain except for the set of the solution discontinuity. When constructing the scheme we use the new coordinates in which the location of boundary layers becomes stationary.