As Kohn-Sham density functional theory (KSDFT) being applied to increasingly more complex materials, the periodic boundary condition associated with supercell approaches also becomes unsuitable for a number of important scenarios. Green's function embedding methods allow a more versatile treatment of complex boundary conditions, and hence provide an attractive alternative to describe complex systems that cannot be easily treated in supercell approaches. In this talk, we first revisit the literature of Green's function embedding methods from a numerical linear algebra perspective. We then propose a new Green's function embedding method called PEXSI-$\Sigma$. As a proof of concept, we demonstrate the accuracy of the PEXSI-$\Sigma$ method for graphene with divacancy and dislocation dipole type of defects using the DFTB+ software package. (Joint with Xiantao Li and Jianfeng Lu)