A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

Abstract

We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) 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( Δx 2( p+1) ) and O( Δx 2 p+1 ) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O( Δx p+2 ) 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→∞. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.