Adaptive p-version of the finite element method for solving boundary value problems for ordinary second-order differential equations

We suggest an adaptive strategy for constructing a hierarchical basis for a p-version of the finite element method used to solve boundary value problems for second-order ordinary differential equations. The choice of the order of an element on each grid interval is based on estimates of the change, in the norm of C, of the approximate solution or the value of the functional to be minimized when increasing the degree of the basis function added on this interval. The results of numerical experiments estimating the method efficiency are given for sample problems whose solutions have singularities of the boundary layer type. We make a comparison with the p-version of the finite element method, which uses a uniform growth of the degree of the basis functions, and with the h-version, which uses uniform grid refinement along with an adaptive grid refinement and coarsening strategy.