Abstract. We obtain the asymptotic estimates of the accuracy of the eigenvalues of the fourth order operator with mixed boundary conditions on the boundary of the rectangle. Knowledge of the main part of eigenvalues error allows us to reasonably specify eigenvalues on a sequence of grids, receive discrete analogs of high accuracy, construct discrete analogs whose eigenvalues give the bilateral approximation to eigenvalues of the original problem.

Keywords: accuracy estimate, elliptic operator, mixed boundary conditions, difference scheme, the principal term of error.