The optimal multilevel Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system

Existence and local-uniqueness theorems for weak solutions of a system consisting of the drift-diffusion-Poisson equations and the Poisson-Boltzmann equation, all with stochastic coefficients, are presented. For the numerical approximation of the expected value of the solution of the system, we develop a multi-level Monte-Carlo (MLMC) finite-element method (FEM) and we analyze its rate of convergence and its computational complexity. This allows to find the optimal choice of discretization parameters. Finally, numerical results show the efficiency of the method. Applications are, among others, noise and fluctuations in nanoscale transistors, in field-effect bio- and gas sensors, and in nanopores.