G. I. Shishkin
Grid approximation of a singularly perturbed one-dimensional heat equation on an unbounded domain excluding a rectangle with sides noncollinear to the axes
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 &egr;-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.
Russian Journal of Numerical Analysis and Mathematical Modelling, Walter de Gruyter
Print ISSN: 0927-6467
Volume: 18, 10/2003
Pages: 429 - 454
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