A challenging problem in quantum field theory is the study of conformal (or nearly-conformal) fixed points occurring in the non-perturbative regime of a quantum field theory. Using radial quantization, computation on curved manifolds is essential. We propose a new approach called Quantum Finite Elements (QFE), an extension of the usual Finite Element Method (FEM) to solving classical PDEs, where renormalization of couplings can play a key role in the restoration of rotational invariance. Some aspects of our approach can be found in earlier work related to Regge calculus and lattice quantum gravity, as well as the random lattice approach of Christ, Friedberg and Lee.