  $L^1$ based optimization is widely used in image denoising, machine learning
and related applications. One of the main features of such approach is that it
naturally provide a sparse structure in the numerical solutions. In this paper,
we study an $L^1$ based mixed DG method for second-order elliptic equations in
the non-divergence form. The elliptic PDE in nondivergence form arises in the
linearization of fully nonlinear PDEs. Due to the nature of the equations,
classical finite element methods based on variational forms can not be employed
directly. In this work, we propose a new optimization scheme coupling the
classical DG framework with recently developed $L^1$ optimization technique.
Convergence analysis in both energy norm and $L^{\infty}$ norm are obtained
under weak regularity assumption. Such $L^1$ models are nondifferentiable and
therefore invalidate traditional gradient methods. Therefore all existing
gradient based solvers are no longer feasible under this setting. To overcome
this difficulty, we characterize solutions of $L^1$ optimization as
fixed-points of proximity equations and utilize matrix splitting technique to
obtain a class of fixed-point proximity algorithms with convergence analysis.
Various numerical examples are displayed to illustrate the numerical solution
has sparse structure with careful choice of the bases of the finite dimensional
spaces. Numerical examples in both smooth and nonsmooth settings are provided
to validate the theoretical results.
