  This paper concerns with numerical approximations of solutions of second
order fully nonlinear partial differential equations (PDEs). A new notion of
weak solutions, called moment solutions, is introduced for second order fully
nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a
constructive method, called vanishing moment method, hence, they can be readily
computed by existing numerical methods such as finite difference, finite
element, spectral Galerkin, and discontinuous Galerkin methods with
"guaranteed" convergence. The main idea of the proposed vanishing moment method
is to approximate a second order fully nonlinear PDE by a higher order, in
particular, a fourth order quasilinear PDE. We show by various numerical
experiments the viability of the proposed vanishing moment method. All our
numerical experiments show the convergence of the vanishing moment method, and
they also show that moment solutions coincide with viscosity solutions whenever
the latter exist.
