INTERNODES is a general method to deal with non-conforming discretizations of second order partial differential equations on regions partitioned into two or several subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interface, the others for the Neumann trace. In this paper we provide several interpretations of the method and we carry out its stability and convergence analysis. In every subdomain the original problem is discretized by the finite element method, using a priori non-matching grids and piecewise polynomials of different degree. Finally, we propose an efficient algorithm for the solution of the corresponding algebraic system.