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In this paper a survey is presented of the use of finite element methods for the simulation of the behaviour of semiconductor devices. Both ordinary and mixed finite element methods are considered. We indicate how the various mathematical models of semiconductor device behaviour can be obtained from the Boltzmann transport equation and the appropriate closing relations. The drift-diffusion and hydrodynamic models are discussed in more detail. Some mathematical properties of the resulting nonlinear systems of partial differential equations are identified, and general considerations regarding their numerical approximations are discussed. Ordinary finite element methods of standard and non-standard type are introduced by means of one-dimensional illustrative examples. Both types of finite element method are then extended to two-dimensional problems and some practical issues regarding the corresponding discrete linear systems are discussed. The possibility of using special non-uniform fitted meshes is noted. Mixed finite element methods of standard and non-standard type are described for both one-and two-dimensional problems. The coefficient matrices of the linear systems corresponding to some methods of non-standard type are monotone. Ordinary and mixed finite element methods of both types are applied to the equations of the stationary drift-diffusion model in two dimensions. Some promising directions for future research are described.

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter H ∈ ( 1 2 , 1). The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.