The periodic first boundary value problem is considered in a band domain for a parabolic-type equation. Highest-order derivatives contain a parameter taking arbitrary values in the half-open interval (0,1], while the equation coefficients and free term have discontinuities of the first kind at a finite number of straight lines parallel to one of the coordinate axes. Lumped sources can also be located at these straight lines. When the parameter tends to zero there arise internal layers in the neighbourhoods of discontinuity lines for the data of the problem. To solve the boundary value problem by using grids condensing at boundary and internal layers, a difference scheme is constructed which converges uniformly in the parameter everywhere in the domain. Efficient numerical methods [11,13] have been devised for approximating sufficiently smooth solutions to boundary value problems. The application of classical methods [2,4,7,[11][12][13][14] to solving problems whose solutions have restricted smoothness leads to a loss in accuracy. In a number of boundary value problems, when the parameter at highest-order derivatives in differential equations tends to zero, the solution to the problem is smooth on the greater part of the domain and it tends to the solution of a non-singular problem. The solution transforms to a finite quantity and its derivatives infinitely increase only on a small part of the domain whose dimensions decrease with the parameter decreasing. Since there is a decrease in smoothness of the solution when the parameter tends to zero, some difficulties arise in solving these problems numerically. Thus, for example, to attain a prescribed accuracy in the case where classical schemes are used on uniform grids, it is necessary to tend the grid step to zero when the parameter tends to zero [7].To solve certain model problems of this type, special difference schemes converging uniformly in the parameter were constructed and justified in [2,7]. These first results rigorously justified for problems with boundary layers were yielded by two different approaches used to construct special numerical methods. In [7], coefficients of difference equations are determined on an arbitrary grid (uniform, for example) in such a way that a uniform (in the parameter) accuracy in approximation is provided. In [2], approximation for traditional difference equations is attained by making some changes in the location of grid nodes. In this way the scheme is adapted for providing convergence uniform in the parameter.Methods of construction of special schemes, which are based on a choice of coefficients of difference equations on arbitrary grids, are widely used (corresponding references and some results can be found, for example, in [4]) as they make it possible to obtain solutions in simple (sufficiently coarse) grid domains. Effective adaptation schemes were constructed (see, for example [1,4,5,7]) for certain boundary value problems (ordinary and partial differential equations which have boundary layers of the exponential type)...

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A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter ε, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates ε-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the ε-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.