  This paper is concerned with developing accurate and efficient numerical
methods for fully nonlinear second order elliptic and parabolic partial
differential equations (PDEs) in multiple spatial dimensions. It presents a
general framework for constructing high order local discontinuous Galerkin
(LDG) methods for approximating viscosity solutions of these fully nonlinear
PDEs. The proposed LDG methods are natural extensions of a narrow-stencil
finite difference framework recently proposed by the authors for approximating
viscosity solutions. The idea of the methodology is to use multiple
approximations of first and second order derivatives as a way to resolve the
potential low regularity of the underlying viscosity solution. Consistency and
generalized monotonicity properties are proposed that ensure the numerical
operator approximates the differential operator. The resulting algebraic system
has several linear equations coupled with only one nonlinear equation that is
monotone in many of its arguments. The structure can be explored to design
nonlinear solvers. This paper also presents and analyzes numerical results for
several numerical test problems in two dimensions which are used to gauge the
accuracy and efficiency of the proposed LDG methods.
