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An efficient implementation of the density-gradient (DG) approach for the finite element and finite difference methods and its application in drift-diffusion (D-D) simulations is described in detail. The new, second-order differential (SOD) scheme is compatible with relatively coarse grids even for large density variations thus applicable to device simulations with complex 3-D geometries. Test simulations of a 1-D metal-oxide semiconductor diode demonstrate that the DG approach discretized using our SOD scheme can be accurately calibrated against Schrödinger-Poisson calculations exhibiting lower discretization error than the previous schemes when using coarse grids and the same results for very fine meshes. 3-D test D-D simulations using the finite element method are performed on two devices: a 10 nm gate length double gate metal-oxide-semiconductor field-effect transistor (MOSFET) and a 40 nm gate length Tri-Gate fin field-effect transistor (FinFET). In 3-D D-D simulations, the SOD scheme is able to converge to physical solutions at high voltages even if the previous schemes fail when using the same mesh and equivalent conditions. The quantum corrected D-D simulations using the SOD scheme also converge with an atomistic mesh used for the 10 nm double gate MOSFET saving computational resources and can be accurately calibrated against the results from non-equilibrium Green's functions approach. Finally, the simulated ID-VG characteristics for the 40 nm gate length Tri-Gate are in an excellent agreement with experimental data.