In the framework of mixed and hybrid finite element methods for diffusion-type and other elliptic differential equations, it is necessary to apply numerical schemes with variables given as values of normal fluxes on the edges (faces) of the elementary cells and values of the scalar-valued function in each cell. In this paper, we propose a general method of constructing finite element approximations on polygonal and polyhedral meshes, whose cells are convex and nonconvex polygonal domains in R2 and polyhedrons in R3. We present a natural way of constructing cell prolongation operators, which makes it possible to easily compute the coefficients of the respective mass matrices. Also, the proposed prolongations satisfy the important requirement that the image of the divergence operator on the extended fields belongs to the set of piecewise constant functions. The latter fact provides direct justification of the well-posedness of the arising discrete problems.