Finite Element Stabilization for Drift-Diffusion/Advection-Diffusion Equations
I've tried my best to look through the relevant suggested similar questions when posting this, and hopefully this contains enough new material to not be considered a duplicate.
I'm currently trying to draw up a semiconductor drift-diffusion simulation framework, and the work that I'm supposed to be working off of, even though it's called 'finite element' in the section heading, is control-volume finite element, or vertex-centered finite volume (depending on which field you come from, one may be more common notation than the other).
As best as I understand it right now, the finite volume method preserve fluxes by construction, and is therefore preferred for implementation of conservation laws. It also allows the Scharfetter-Gummel Finite Box method to be implemented for current densities across the finite volume elements. The SG-FB method is a generalization of the original SG formulation, where the finite differences between carrier densities (the $\nabla n$ term below) are calculated using a special exponential smoothing scheme. This is done because $n$ can often change discontinuously between two points in space, and the SG method is a stable difference scheme.
However, I would like to implement this in finite elements, not volume, almost entirely because of the libraries available for FEM (deal.II, libMesh, FEniCS, ...). 232
The relevant current density equations for electrons are below (these, combined with an identical equation for holes, and a nonlinear Poisson equation, make up the system)
$$ \frac{\partial n}{\partial t} = \frac{1}{q} \nabla \cdot \mathbf{J_n} + G_n - R_n $$
$$ \mathbf{J_n} = qD_n\nabla n - q\mu_ nn\nabla \phi $$
For the questions:
1. Are there any flux/mass-conserving schemes in FEM that have properties similar/identical to the flux/mass-conserving properties of FVM? I keep seeing in a few places that discontinuous-Galerkin conserves flux/mass, but I can't quite get a consensus on it.
2. What would be appropriate stabilization scheme(s) to apply to this system, given the rapid fluctuations in $n$ (and $p$, identically, for the hole continuity equation)? I keep reading about streamline-upwind Petrov-Galerkin, Galerkin least Squares, and shock-capturing terms, but this seems to be a generally open question.
3. Given that I keep seeing shock capturing terms in my research, is it reasonable to claim that rapid, discontinuous changes in carrier density are identical to 'static' shock waves/fronts?
I'm still quite new to these equations, having dealt with them only for the past few months. I'd be happy to provide any more information, if required. 232