Cryogenic nanoscale semiconductor devices are crucial for a wide variety of applications. The accurate design of such devices involve solving their electrostatics from fundamental semiconductor equations at low temperatures. We employ COMSOL to model a prototype cryogenic nanostructure, that can be used to readout the spin of a single electron in silicon, for quantum computing applications. By achieving convergence down to 15 Kelvin (K), we provide a guideline of techniques that aid to enhance convergence at cryogenic temperatures. We further compare the device electrostatics at different temperatures, which aids us to estimate the accuracy with temperature.