  We consider the construction of locally conservative fluxes by means of a
simple post-processing technique obtained from the finite element solutions of
advection diffusion equations. It is known that a naive calculation of fluxes
from these solutions yields non-conservative fluxes. We consider two finite
element methods: the usual continuous Galerkin finite element (CGFEM) for
solving non dominating advection diffusion equations and the streamline
upwind/Petrov-Galerkin (SUPG) for solving advection dominated problems. We then
describe the post-processing technique for constructing conservative fluxes
from the numerical solutions of the general variational formulation. The
post-processing technique requires solving an auxiliary Neumann boundary value
problem on each element independently and it produces a locally conservative
flux on a vertex centered dual mesh relative to the finite element mesh. We
provide a convergence analysis for the post-processing technique. Performance
of the technique and the convergence behavior are demonstrated through
numerical examples including a set of test problems for advection diffusion
equations, advection dominated equations, and drift-diffusion equations.
