DG discretizations for compressible flows: Adjoint consistency analysis, error estimation and adaptivity

Abstract

<strong>1. Adjoint consistency</strong> - 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 J(.). If the primal and adjoint solutions are sufficiently smooth, the order of convergence in J for an adjoint consistent discretization is twice the order of an adjoint inconsistent discretization (order doubling). In this talk we provide a general framework for analyzing the adjoint consistency of DG discretizations. We collect several conclusions which can be drawn from analyzing the adjoint consistency property of DG discretizations of the linear advection equation, Poisson's equation and the compressible Euler and Navier-Stokes equations. <strong>2. Error estimation and adaptivity:</strong> We give some examples of applying an <em>a posteriori</em> error estimation and adjoint based refinement to inviscid and viscous laminar sub-, trans- and supersonic compressible flows also in combination with anisotropic mesh refinement.