Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations
SIAM J. Numer. Anal. Volume 45, Issue 6, pp. 2671-2696 (2007)
http://dx.doi.org/10.1137/060665117

This paper gives a notion of consistent adjoint discretizations for DG methods. However the definition seems to be restricted to smooth solutions only. For the case of a conservation law, they show that boundary conditions of the primal problem must be appropriate for the corresponding adjoint discretization to be consistent. The DG method requires the use of a numerical flux function. It is common among many CFD codes to use the same numerical flux function for computing the boundary fluxes, with the help of ghost data which incorporates the boundary conditions. The authors show that this does not lead to consistent adjoint discretizations. One must use a boundary flux which is consistent with the exact boundary conditions. Without knowing this, I have always followed the correct approach, i.e., I never use the numerical flux function for boundary flux computation. In the case of Euler equations, the boundary flux contains only the pressure contribution. Hence one can extrapolate the pressure from the interior cell upto the boundary face, and then compute the flux using the exact analytical definition of flux, i.e.,



for the 2-D case.