We are interested in the approximation of partial differential equations on domains decomposed into two (or several) subdomains featuring non-conforming interfaces. The non-conformity may be due to different meshes and/or different polynomial degrees used from the two sides, or even to a geometrical mismatch. Across each interface, one subdomain is identified as master and the other as slave. We consider Galerkin methods for the discretization (such as finite element or spectral element methods) that make use of two interpolants for transferring information across the interface: one from master to slave and another one from slave to master. The former is used to ensure continuity of the primal variable (the problem solution), while the latter that of the dual variable (the normal flux). In particular, since the dual variable is expressed in weak form, we first compute a strong representation of the dual variable from the slave side, then interpolate it, transform the interpolated quantity back into weak form and eventually assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. The proposed method is named INTERNODES (INTERpolation for NOnconforming DEcompositionS): it can be regarded as an alternative to the mortar element method and it is simpler to implement in a numerical code. We show on two dimension al problems that by using the Lagrange interpolation we obtain at least as good convergence results as with the mortar element method with any order of polynomials. When using low order polynomials, the radial-basis interpolant leads to the same convergence properties as the Lagrange interpolant. We conclude with a comparison between INTERNODES and a standard conforming approximation in a three dimensional case.