Symmetric Interior Penalty DG Methods for the Compressible Navier{Stokes Equations I: Method formulation

Abstract. In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin nite element method to the numerical approxima-tion of the compressible Navier{Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the nite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical nite element meshes specically tailored to the computation of the tar-get functional of interest, as well as providing ecient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented. Key words. Discontinuous Galerkin methods, a posteriori error estimation, adaptivity, com-pressible Navier{Stokes equations 1