An Adjoint Consistency Analysis for a Class of Hybrid Mixed Methods

Hybrid methods represent a classic discretization paradigm for elliptic equations. More recently, hybrid methods have been formulated for convection-diffusion problems, in particular compressible fluid flow. In [25], we have introduced a hybrid mixed method for the compressible Navier-Stokes equations as a combination of a hybridized DG scheme for the convective terms, and an H(div,Ω)-method for the diffusive part. Since hybrid methods are based on Galerkin’s principle, the adjoint of a given hybrid discretization may be used for PDE-constraint optimal control problems, or error estimation, provided that the discretization is adjoint consistent. In the present paper, we extend the adjoint consistency analysis, previously reported for many DG schemes to the more complex hybrid methods. We prove adjoint consistency for a class of Hybrid Mixed schemes, which includes the hybridized DG schemes proposed by [19], as well as our recently proposed method ([25]). Hybrid Mixed discretizations, Hybridized Discontinuous Galerkin discretizations, Adjoint Consistency, compressible Navier-Stokes equations