{'fraction_non_alphanumeric': 0.02164644145621515, 'fraction_numerical': 0.004919645785503444, 'mean_word_length': 5.39412997903564, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Grid approximation of singularly perturbed boundary value problems with a regular boundary layer

The Dirichlet problem is considered for an equation of elliptic type. Highest-order derivatives of the equation contain a parameter taking arbitrary values in the half-open interval (0,1]; with the parameter value equal to zero, the differential equation degenerates into a zero-order equation. The domain boundary is assumed to be smooth. When the parameter tends to zero there arises in a neighbourhood of the boundary a regular boundary layer described with an ordinary differential equation. For an equation with mixed derivatives, methods based on fitting schemes for ordinary differential equations are suggested for constructing schemes converging uniformly in the parameter. The schemes constructed are monotonic and converge uniformly in the parameter with a first-order accuracy. Boundary value problems for elliptic equations with small parameters at highest-order derivatives are referred to singularly perturbed problems. We consider boundary value problems for an elliptic equation degenerating into a zero-order equation. In the case of smooth data of problem (the coefficients and the free term), if the parameter tends to zero, boundary layers arise in a neighbourhood of the boundary. These boundary layers are regular and described with ordinary differential equations if the boundary of the domain is smooth [21]. The methods of construction of difference schemes for singularly perturbed ordinary differential equations are sufficiently well developed at present (see, for example, [1,3,8] and the references to [3,4]). Therefore, it would be interesting to make use of these results for equations with partial derivatives with a regular boundary layer. In this paper, when designing schemes we exploit fitting schemes constructed on the basis of exact two-point schemes [13-15]. The following idea underlies the construction of the scheme for the boundary value problem. In the case of the regular boundary layer, the partial derivatives of the solution to the boundary value problem in the directions tangent to the domain boundary are bounded uniformly in the parameter. The differential equation with partial derivatives can therefore be considered as a singularly perturbed ordinary differential equation in a variable orthogonal to the boundary of the domain. The free term of this ordinary differential equation, which contains directional derivatives tangent to the boundary is bounded uniformly in the parameter. For an ordinary differential equation with a special right-hand side we construct by using techniques of exact difference schemes [12] functional-difference equations whose solution coincides with that to the boundary value problem. The difference operator of such equations is three-point and monotonic. Using this difference operator we construct a monotonic scheme (a fitting scheme) for the ordinary differential equation. In the case of boundary value problems on the η-dimensional layer, to approximate