  Viable non-perturbative methods for lattice quantum field theories on curved
manifolds are difficult. By adapting features from the traditional finite
element methods (FEM) and Regge Calculus, a new simplicial lattice Quantum
Finite Element (QFE) Lagrangian is constructed for fields on a smooth Riemann
manifold. To reach the continuum limit additional counter terms must be
constructed to cancel the ultraviolet distortions. This is tested by the
comparison of phi 4-th theory at the Wilson-Fisher fixed point with the exact
Ising (c =1/2) CFT on a 2D Riemann sphere. The Dirac equation is also
constructed on a simplicial lattice approximation to a Riemann manifold by
introducing a lattice vierbein and spin connection on each link. Convergence of
the QFE Dirac equation is tested against the exact solution for the 2D Riemann
sphere. Future directions and applications to Conformal Field Theories are
suggested.
