  This paper studies mixed finite element approximations of the viscosity
solution to the Dirichlet problem for the fully nonlinear Monge-Amp\`ere
equation $\det(D^2u^0)=f$ based on the vanishing moment method which was
proposed recently by the authors in \cite{Feng2}. In this approach, the second
order fully nonlinear Monge-Amp\`ere equation is approximated by the fourth
order quasilinear equation $-\epsilon\Delta^2 u^\epsilon + \det{D^2u^\epsilon}
=f$. It was proved in \cite{Feng1} that the solution $u^\epsilon$ converges to
the unique convex viscosity solution $u^0$ of the Dirichlet problem for the
Monge-Amp\`ere equation. This result then opens a door for constructing
convergent finite element methods for the fully nonlinear second order
equations, a task which has been impracticable before. The goal of this paper
is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite
element methods for approximating the solution $u^\epsilon$ of the regularized
fourth order problem, which computes simultaneously $u^\vepsi$ and the moment
tensor $\sigma^\vepsi:=D^2u^\epsilon$. Second, we derive error estimates, which
track explicitly the dependence of the error constants on the parameter
$\vepsi$, for the errors $u^\epsilon-u^\epsilon_h$ and
$\sigma^\vepsi-\sigma_h^\vepsi$. Finally, we present a detailed numerical study
on the rates of convergence in terms of powers of $\vepsi$ for the error
$u^0-u_h^\vepsi$ and $\sigma^\vepsi-\sigma_h^\vepsi$, and numerically examine
what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve
these rates.
