Error Estimates For Local Discontinuous Galerkin Methods With Nonsmooth Initial Data

In this paper, we discuss the LDG method for one-dimensional linear constant coeffi-cient convection-diffusion equations with periodic boundary conditions and nonsmooth initial data, estimate its L2 error and improve the error bound for large time. Inspired by works of error estimates for classical finite element methods with nonsmooth initial data, we first obtain error estimates for initial data in Hk+1 (k> 0 is the degree of polynomials of the finite element space) by means of elliptic projection and energy ar-gument and then obtain error estimates for arbitrary data in L2 using properties of error operators and the Fourier theory. It is Pouicare’s inequality that we use to improve the error bound for large time. Numerical experiments demonstrate that our error bound exactly describes the order of accuracy, how the error changes with time and how the smooth degree affects the error.