Construction of basis functions for finite element methods in a Hilbert space

The present work is devoted to construction of the optimal interpolation formula exact for trigonometric functions sin($\omega$x) and cos($\omega$x). Here the analytical representations of the coefficients of the optimal interpolation formula in a certain Hilbert space are obtained using the discrete analogue of the differential operator. Taking the coefficients of the optimal interpolation formula as basis functions, in the finite element methods the boundary value problems for ordinary differential equations of the second order are approximately solved. In particular, it is shown that the coefficients of the optimal interpolation formula can serve as a set of effective basis functions. Approximate solutions of the differential equations are compared using the constructed basis functions and known basis functions. In particular, we have obtained numerical results for the cases when the numbers of basis functions are 6 and 11. In both cases, we have got that the accuracy of the approximate solution to the boundary value problems for second-order ordinary differential equations found using our basis functions is higher than the accuracy of the approximate solution found using known basis functions. It is proven that the accuracy of the approximate solution increases with increasing the number of basis functions.