An optimal estimate for the Local Discontinuous Galerkin method

L-2 error estimates for the Local Discontinuous Galerkin (LDG) method have been theoretically proven for linear convection diffusion problems and periodic boundary conditions. It has been proven that when polynomials of degree k are used, the LDG method has a suboptimal order of convergence k. However, numerical experiments show that under a suitable choice of the numerical flux, higher order of convergence can be achieved. In this paper, we consider Dirichlet boundary conditions and we show that the LDG method has an optimal order of convergence k + 1.