Superconvergence of Local Discontinuous Galerkin method for   one-dimensional linear parabolic equations
  In this paper, we study superconvergence properties of the local discontinuous Galerkin method for one-dimensional linear parabolic equations when alternating fluxes are used. We prove, for any polynomial degree $k$, that the numerical fluxes converge at a rate of $2k+1$ (or $2k+1/2$) for all mesh nodes and the domain average under some suitable initial discretization.   We further prove a $k+1$th superconvergence rate for the derivative approximation and a $k+2$th superconvergence rate for the function value approximation at the Radau points.   Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp. 
Waixiang Cao and Zhimin Zhang
math.NA