THE CONSTRUCTION of schemes of the finite-element method of an arbitrary order of accuracy for second-order elliptic equations in arbitrary sufficiently smooth regions is accomplished in such a way that they are spectrally equivalent to the simplest finite difference approximations of the Laplace equation on a uniform orthogonal mesh. This enables us to construct efficient iterative methods for solving these schemes. Iterative methods on sequences of discharging networks, leading to optimal-order estimates of the number of arithmetic operations are also considered. The results obtained lead to the development of economy of schemes of high orders of accuracy.