  We present a method for defining a lattice realization of the $\phi^4$
quantum field theory on a simplicial complex in order to enable numerical
computation on a general Riemann manifold. The procedure begins with adopting
methods from traditional Regge Calculus (RC) and finite element methods (FEM)
plus the addition of ultraviolet counter terms required to reach the
renormalized field theory in the continuum limit. The construction is tested
numerically for the two-dimensional $\phi^4$ scalar field theory on the Riemann
two-sphere, $\mathbb{S}^2$, in comparison with the exact solutions to the
two-dimensional Ising conformal field theory (CFT). Numerical results for the
Binder cumulants (up to 12th order) and the two- and four-point correlation
functions are in agreement with the exact $c = 1/2$ CFT solutions.
