Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations

This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in $L^2$ as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier–Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.