Using a functional-discrete approach, three-point difference schemes of arbitrary order of accuracy are constructed for solving the Dirichlet problem for second-order ordinary differential equations (ODE) with a small parameter multiplying the leading derivative. The uniform convergence of the schemes with respect to the small parameter is proved, and a recursive algorithm for their realization is constructed. Bibliography:4 titles.

Homogeneous three-point finite-difference schemes of arbitrary order of accuracy for a second-order Dirichlet problem with generalized coefficients and generalized right-hand side are constructed and validated. Bibliography: 8 titles.

In a rectangular domain we construct a grid scheme by applying the operators of exact difference schemes. We study an estimate of the rate of convergence of the grid scheme in the grid norm L2(ω). It is shown that in the case when the solution of the differential problem belongs to the space W 2 k (Ω), k ∈ (3/2,2] the order of precision of the proposed scheme is O(hk−3/2), and in the linear case it is O(hk).