  We extend a new finite element code, Einstein PHG (iPHG), to solve the
evolution part of Einstein equations in first-order GH formalism. This paper is
the third one of a systematic investigation of applying adaptive finite element
method to the Einstein equations, especially binary compact objects
simulations. The primary motivation of this work is to evolve black holes for
the first time utilizing a continuous Galerkin finite element method on
unstructured (tetrahedral) mesh. We test our code by evolving a nonlinear
scalar wave equation. It works well and runs stably with both reflect and
radiative boundary conditions. Then we use iPHG to simulate the full
three-dimensional spacetime of a single black hole. We find that the filter
used to dealt with aliasing error is a crucial ingredient for numerical
stability. For simplicity, we impose the "freezing" ingoing characteristic
fields condition in weak form at the outer boundary. Our simulations show both
the convergence and stability.
