In this paper, we present the optimal L 2 -error estimate of O(h k+1 ) for polynomial elements of degree k of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so

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... hat trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of k + 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on Cartesian grids. A collection of projection errors in both one-and multi-dimensional cases is established.