  Kohn-Sham density functional theory is one of the most widely used electronic
structure theories. In the pseudopotential framework, uniform discretization of
the Kohn-Sham Hamiltonian generally results in a large number of basis
functions per atom in order to resolve the rapid oscillations of the Kohn-Sham
orbitals around the nuclei. Previous attempts to reduce the number of basis
functions per atom include the usage of atomic orbitals and similar objects,
but the atomic orbitals generally require fine tuning in order to reach high
accuracy. We present a novel discretization scheme that adaptively and
systematically builds the rapid oscillations of the Kohn-Sham orbitals around
the nuclei as well as environmental effects into the basis functions. The
resulting basis functions are localized in the real space, and are
discontinuous in the global domain. The continuous Kohn-Sham orbitals and the
electron density are evaluated from the discontinuous basis functions using the
discontinuous Galerkin (DG) framework. Our method is implemented in parallel
and the current implementation is able to handle systems with at least
thousands of atoms. Numerical examples indicate that our method can reach very
high accuracy (less than 1meV) with a very small number ($4\sim 40$) of basis
functions per atom.
