In this paper we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven.

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We present an analysis of transonic solutions of the steady state 1-dimensional unipolar hydrodynamic model for semiconductors in the isoentropic case. The approach is based on construction of the orbits of the system in the electron density-electric field phase plane and on representation of discontinuous solutions of the hydrodynamic boundary value problem by a union of trajectory pieces. These pieces are related by shocks obeying jump and entropy conditions. A continuation argument in the length of the semiconductor device under consideration is applied to construct a continuum of sub- and transonic solutions, which contains at least one solution for every positive length. We also present numerical results illustrating the various possible solution profiles. For this we use a regularization of the problem, adding artificial diffusion to obtain singularly perturbed problems which are then solved numerically using continuation in the regularization parameter.

We apply Wigner-transform techniques to the analysis of difference methods for Schrödinger-type equations in the case of a small Planck constant. In this way we are able to obtain sharp conditions on the spatialtemporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not.Numerical test examples are presented to help interpret the theory. Mathematics Subject Classification (1991): 65M12

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.1991 Mathematics Subject Classification. 65N30, 65N12, 65N15, 35J05.The dates will be set by the publisher.

The energy-transport models describe the flow of electrons through a semiconductor crystal, influenced by diffusive, electrical and thermal effects. They consist of the continuity equations for the mass and the energy, coupled to Poisson's equation for the electric potential. These models can be derived from the semiconductor Boltzmann equation. This paper consists of two parts. The first part concerns with the modelling of the energy-transport system. The diffusion coefficients and the energy relaxation term are computed in terms of the electron density and temperature, under the assumptions of nondegenerate statistics and non-parabolic band diagrams. The equations can be rewritten in a drift-diffusion formulation which is used for the numerical discretization. In the second part, the stationary energy-transport equations are discretized using the exponential fitting mixed finite element method in one space dimension. Numerical simulations of a ballistic diode are performed.

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Abstract. We apply Wigner transform techniques to the analysis of the Dufort-Frankel difference scheme for the Schrödinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort-Frankel scheme to other difference schemes for the Schrödinger equation.