In this article we propose a new symmetric version of the
interior penalty discontinuous Galerkin finite element
method for the numerical approximation
of the compressible Navier-Stokes equations. Here, particular
emphasis is devoted to the construction of an optimal numerical
method for the evaluation of certain target functionals of practical
interest, such as the lift and drag coefficients of a body immersed in a
viscous fluid. With this in mind, the key ingredients in the
construction of the method include: (i) An adjoint consistent imposition of
the boundary conditions; (ii) An adjoint consistent reformulation of the
underlying target functional of practical interest; (iii) Design of appropriate
interior--penalty stabilization terms. Numerical experiments presented
within this article clearly indicate the optimality of the proposed
method when the error is measured in terms of both the L2-norm, as well as
for certain target functionals. Computational comparisons with
other discontinuous Galerkin schemes proposed in the literature,
including the second scheme
of Bassi and Rebay, the standard SIPG method
outlined in [Hartmann,Houston-2006], and an NIPG variant of the new scheme
will be undertaken
