LAUSR

We provide a framework for the analysis of the direct discontinuous Galerkin (DDG) methods for multi-dimensional convection-diffusion equations subject to various boundary conditions. A key tool is the global projection constructed by the DDG scheme applied to an associated elliptic problem. Such projection is well-defined for a class of diffusive flux parameters, and the optimal projection error in L is obtained with an arbitrary locally regular partition of the domain and for an arbitrary degree of polynomials. This results in the optimal L error for the DDG method to the elliptic problem, and further leading to the optimal L error for the DDG method to the convection-diffusion problem.