  Achieving self-consistent convergence for conventional effective-mass
approaches at ultralow temperatures (below 4.2 K) is a challenging task, which
mostly lies in the discontinuities in material properties (e.g.,
effective-mass, electron affinity, dielectric constant). In this article, we
develop a novel self consistent approach based on cell centered Finite Volume
discretization of the Sturm-Liouville form of the effective mass Schrodinger
equation and generalized Poisson's equation (FV-SP). We apply this approach to
simulate the one dimensional electron gas (1DEG) formed at the Si-SiO2
interface via a top gate. We find excellent self-consistent convergence from
high to extreme low (as low as 50 mK) temperatures. We further examine the
solidity of FV-SP method by changing external variables such as the
electrochemical potential and the accumulative top gate voltage. Our approach
allows for counting electron-electron interactions. Our results demonstrate
that FV-SP approach is a powerful tool to solve effective mass Hamiltonians.
