Derivation of an adjoint consistent discontinuous Galerkin discretization of the compressible Euler equations

Abstract

Adjoint consistency - in addition to consistency - is the key requirement for discontinuous Galerkin (DG) discretizations to be of optimal order in L² as well as measured in terms of target functionals. Furthermore, adjoint consistency is closely related to the smoothness of discrete adjoint solutions. Whereas adjoint solutions based on the (non-adjoint-consistent) NIPG method are discontinuous between element interfaces, where the jumps in the adjoint solutions even persist as the mesh is refined, the adjoint solutions based on the (adjoint-consistent) SIPG method are essentially continuous. Furthermore an appropriate modification of a specific target functional for the Laplace equation is required for recovering an adjoint consistent discretization. Recently, a specific discretization of the boundary fluxes and the target functional has been proposed to recover an adjoint consistent DG discretization of the compressible Euler equations. In this paper, we provide a general framework for analyzing adjoint consistency of DG discretizations and introduce so-called consistent modifications of target functionals. Whereas the standard DG discretization of compressible Euler equations is not adjoint consistent, we use the outlined framework to derive the adjoint consistent discretization of the compressible Euler equations. Numerical experiments demonstrate the effect of adjoint consistency on the smoothness of discrete adjoint solutions and on the <em>a posteriori</em> error estimation on adaptively refined meshes. The developed framework is particularly useful for deriving adjoint consistent discretizations of more complex nonlinear problems.