Аннотация:We construct a variant of the finite element method with piecewise-linear basis functions for finding generalized solutions to boundary value problems for systems of second-order linear ordinary differential equations with a symmetric operator. This method uses a sequence of adaptively refined meshes for obtaining a solution with the desired accuracy. The method does not require any information concerning the locations of singularities of the solution and allows one to start calculations on a grid which contains only one internal node. For problems whose solutions possess an exponential singularity, the proposed adaptive method allows the number of nodes and the calculation time to be reduced by a factor of several tens as compared to the finite element method which uses the strategy of uniform mesh refinement.