* A Discontinuous Galerkin Approach to Wigner's Equation*

**Richard Sharp**

University of Texas

email: `rsharp@math.utexas.edu`

**Abstract**: The Wigner equation connects an isolated quantum
system to external effects, which provides an appropriate setting for quantum
device simulation. There are relatively few numerical simulations of Wigner's
equation. Discrete and operator splitting methods have been produced, but each
faces various limitations. We propose a discontinuous-Galerkin (DG) approach
based on a non-polynomial approximation space. Yuan and Shu introduced the use
of such approximation spaces to improve DG performance by reducing the
complexity of the representation of the solution through careful selection of
the basis. We will choose a basis based on numerical and physical
considerations. Numerically, the goal is to use a small number of basis
functions on a coarse mesh. A physically ideal situation exists (i.e. the
solution over a mesh with one cell corresponding to the domain of the
problem), but is not implementable. The practical approach will be to select
basis functions that are both easy to implement and that provide economic
representations of the physical solution. Examples include trigonometric
functions in a region where the solution is oscillatory, or the eigenstates of
an appropriate operator restricted to the mesh.