Asymptotic Formulas for Precision of Discrete Spectral Problems

We find principal terms in the power expansion, with respect to the step of a square grid, of the eigenvalue error for a discrete analogue of spectral problems for elliptic operators of the second and fourth order. We use the compactness of a bounded set in a Hilbert space, which gives the mean convergence of piecewise-constant fillings of grid eigenfunctions and the weak convergence of these fillings for difference derivatives. This, in turn, allows one to prove that eigenfunctions of the initial problems belong to the corresponding Sobolev spaces.