Discontinuous Petrov-Galerkin Method for Compressible Viscous Flows in Three Dimensions

The Discontinuous Petrov-Galerkin (DPG) method allows one to construct stable finite element schemes for some classes of singularily perturbed problems like, for instance, convection-dominated diffusion. The central ingredient of the method is a special weak formulation characterized by a relaxed interelement continuity of approximation. It satisfies the inf-sup stability condition with the stability constant independent of the small parameter of perturbation. This technology was applied by Chan et al. in [1] to develop a scheme for simulation of the two-dimensional compressible Navier-Stokes equations. In this paper we extend that work to three dimensions. We present the necessary modifications, the details of the functional setting, the discrete trial and test finite element spaces, and the final form of the algorithm. The issues of a posteriori error estimation and h-adaptivity of finite element meshes are addressed. The technique is illustrated with a few preliminary numerical examples.