In expanding numerical modeling for electronic and optoelectronic devices from a single dimension to multiple dimensions, a large increase in machine storage space is required. Solution approaches based on relaxation techniques are typically used to minimize this increase, but they can be slow to converge. Presented is an adaption of Brandt's multilevel acceleration scheme for control volume discretizations coupled with solvers based on either Stone's strongly implicit method or the Gauss-Siedel (G-S) method to overcome this speed and storage space problem. This approach is demonstrated by solving Poisson's equation in a two-dimensional amorphous silicon thin-film transistor structure. The structure has a generalized density of states function whose occupancy is computed using nonzero degree Kelvin Fermi-Dirac statistics. It is shown that the use of the multilevel acceleration algorithm gives more than an order of magnitude increase in the asymptotic rate of convergence for the potential distribution in this thin-film transistor. Numerical results of the analysis are presented.>