Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convectiondominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection-dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time-integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint-weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade-off for various approximations of the fine-space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest-neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright 2011 John Wiley & Sons, Ltd.