  We extend the finite element method introduced by Lakkis and Pryer [2011] to
approximate the solution of second order elliptic problems in nonvariational
form to incorporate the discontinuous Galerkin (DG) framework. This is done by
viewing the NVFEM as a mixed method whereby the finite element Hessian is an
auxiliary variable in the formulation. Representing the finite element Hessian
in a discontinuous setting yields a linear system of the same size and having
the same sparsity pattern of the compact DG methods for variational elliptic
problems. Furthermore, the system matrix is very easy to assemble, Thus this
approach greatly reduces the computational complexity of the discretisation
compared to the continuous approach.
  We conduct a stability and consistency analysis making use of the unified
framework set out in Arnold et. al. [2001]. We also give an apriori analysis of
the method. The analysis applies to any consistent representation of the finite
element Hessian, thus is applicable to the previous works making use of
continuous Galerkin approximations.
