  This paper develops interior penalty discontinuous Galerkin (IP-DG) methods
to approximate $W^{2,p}$ strong solutions of second order linear elliptic
partial differential equations (PDEs) in non-divergence form with continuous
coefficients. The proposed IP-DG methods are closely related to the IP-DG
methods for advection-diffusion equations, and they are easy to implement on
existing standard IP-DG software platforms. It is proved that the proposed
IP-DG methods have unique solutions and converge with optimal rate to the
$W^{2,p}$ strong solution in a discrete $W^{2,p}$-norm. The crux of the
analysis is to establish a DG discrete counterpart of the Calderon-Zygmund
estimate and to adapt a freezing coefficient technique used for the PDE
analysis at the discrete level. As a byproduct of our analysis, we also
establish broken $W^{1,p}$-norm error estimates for IP-DG approximations of
constant coefficient elliptic PDEs. Numerical experiments are provided to gauge
the performance of the proposed IP-DG methods and to validate the theoretical
convergence results.
