Convergent Semidiscretization of a Nonlinear Fourth Order Parabolic System

Abstract

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. The resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions as long as the lattice temperature is sufficiently large. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution the rate of convergence proves to be optimal.