In this paper we propose a rescaled localized radial basis function (RL-RBF) interpolation method, based on the use of compactly supported radial basis functions. Starting from the classical RBF interpolation technique, we introduce a rescaling that allows for exact interpolation of constant fields between nonconforming meshes without the use of an extra polynomial term. We also present two-dimensional and three-dimensional numerical examples on arbitrary finite element meshes to show that the RL-RBF interpolation leads to accurate results, fast evaluation, and easy parallelization of the algorithm. All the computations are carried out using the open source finite element library LifeV.