  This paper is concerned with developing accurate and efficient discontinuous
Galerkin methods for fully nonlinear second order elliptic and parabolic
partial differential equations (PDEs) in the case of one spatial dimension. The
primary goal of the paper to develop a general framework for constructing high
order local discontinuous Galerkin (LDG) methods for approximating viscosity
solutions of these fully nonlinear PDEs which are merely continuous functions
by definition. In order to capture discontinuities of the first order
derivative $u_x$ of the solution $u$, two independent functions $q_1$ and $q_2$
are introduced to approximate one-sided derivatives of $u$. Similarly, to
capture the discontinuities of the second order derivative $u_{xx}$, four
independent functions $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are used to
approximate one-sided derivatives of $q_1$ and $q_2$. The proposed LDG
framework, which is based on a nonstandard mixed formulation of the underlying
PDE, embeds a given fully nonlinear problem into a mostly linear system of
equations where the given nonlinear differential operator must be replaced by a
numerical operator which allows multiple value inputs of the first and second
order derivatives $u_x$ and $u_{xx}$. An easy to verify criterion for
constructing "good" numerical operators is also proposed. It consists of a
consistency and a generalized monotonicity. To ensure such a generalized
monotonicity, the crux of the construction is to introduce the numerical moment
in the numerical operator. The proposed framework extends a companion finite
difference framework developed by the authors in [9] and allows for the
approximation of fully nonlinear PDEs using high order polynomials and
non-uniform meshes.
