  We present a real-space formulation for coarse-graining Kohn-Sham Density
Functional Theory that significantly speeds up the analysis of material defects
without appreciable loss of accuracy. The approximation scheme consists of two
steps. First, we develop a linear-scaling method that enables the direct
evaluation of the electron density without the need to evaluate individual
orbitals. We achieve this by performing Gauss quadrature over the spectrum of
the linearized Hamiltonian operator appearing in each iteration of the
self-consistent field method. Building on the linear-scaling method, we
introduce a spatial approximation scheme resulting in a coarse-grained Density
Functional Theory. The spatial approximation is adapted so as to furnish fine
resolution where necessary and to coarsen elsewhere. This coarse-graining step
enables the analysis of defects at a fraction of the original computational
cost, without any significant loss of accuracy. Furthermore, we show that the
coarse-grained solutions are convergent with respect to the spatial
approximation. We illustrate the scope, versatility, efficiency and accuracy of
the scheme by means of selected examples.
