A Posteriori Error Estimation for Discontinuous Galerkin Solutions of Hyperbolic Problems

We analyze the spatial discretization errors associated with solutions of onedimensional hyperbolic conservation laws by discontinuous Galerkin methods in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p+1 on each element. We also prove that the local and global discretization errors are O( x) and O( x) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O( x) at the remaining roots of Radau polynomial of degree p+1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p+1 also holds for smooth solutions as p!1. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are e ective for linear and nonlinear conservation laws in regions where solutions are smooth.