On Adaptive Grid Approximations in the Weight Norm

The purpose of this paper is to develop an algorithm construction of adaptive variation-grid methods for classes of one-dimensional boundary value problems of the second order. Classes of non-degenerate problems are considered, as well as classes of problems with weak and strong degeneracy. The results obtained are suitable for learning computer systems designed to solve problems of the aforementioned classes. To achieve the set goals, the corresponding approximation theorems are established with degeneration. Ways of adaptive choice approximation space at a variation-grid method in a one-dimensional boundary value problem are considered. The locality of the approximation is substantially used. The considerations are reduced to an iterative process, while building an adaptive grid. Numerical examples illustrating the effectiveness of the proposed approach are given.