  This paper develops and analyzes finite element Galerkin and spectral
Galerkin methods for approximating viscosity solutions of the fully nonlinear
Monge-Amp\`ere equation $\det(D^2u^0)=f$ based on the vanishing moment method
which was developed by the authors in \cite{Feng2,Feng1}. In this approach, the
Monge-Amp\`ere equation is approximated by the fourth order quasilinear
equation $-\epsilon\Delta^2 u^\epsilon + \det{D^2u^\epsilon} =f$ accompanied by
appropriate boundary conditions. This new approach allows one to construct
convergent Galerkin numerical methods for the fully nonlinear Monge-Amp\`ere
equation, a task which has been impracticable before. In this paper, we first
develop some finite element and spectral Galerkin methods for approximating the
solution $u^\epsilon$ of the regularized fourth order problem. We then derive
optimal order error estimates for the proposed numerical methods. In
particular, we track explicitly the dependence of the error bounds on the
parameter $\vepsi$, for the error $u^\epsilon-u^\epsilon_h$. Finally, using the
Aygris finite element method as an example, we present a detailed numerical
study of the rates of convergence in terms of powers of $\vepsi$ for the error
$u^0-u_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in
relation to $\vepsi$ in order to achieve these rates.
