Analytical and numerical investigation of the influence of artificial viscosity in discontinuous galerkin methods on an adjoint-based error estimator

Abstract

Recently, it has been observed that the standard approximation to the dual solution in a scalar finite difference context can actually fail if the underlying forward solution is not smooth (Giles and Ulbrich, Convergence of linearised and adjoint approximations for discontinuous solutions of conservation laws. Technical Report, TU Darmstadt and oxford university, oxford, 2008). To circumvent this, it has been proposed to over-refine shock structures of the primal solution. We give evidence that this is also the case in the discontinuous Galerkin approach for the one-dimensional Euler equations if one explicitly adds diffusion. Despite this, on the first sight very negative result, we demonstrate that, if using the dual solution only for adaptation purposes, a special treatment seems not to be necessary to get good convergence in terms of a target functional. © 2011 Springer-Verlag Berlin Heidelberg.