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The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of ongoing work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations, the development of a Gas-Kinetic (BGK) inter-cell flux function implementation, and the extension to three dimensions are given.

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A combined h and p multigrid solution strategy is developed for high-order Discontinuous Galerkin discretizations of the three-dimensional Euler equations. This solver is used to compute inviscid compressible flow over realistic three-dimensional aerodynamic configurations, and the performance of the solver in terms of convergence efficiency and parallel scalability is investigated. The hp multigrid solver is found to deliver nearly optimal convergence rates, which are insensitive to the discretization order p, and to the mesh resolution h. The solver is also shown to scale well on massively parallel computer architectures, demonstrating good scalability up to 2008 processors of the NASA Columbia Supercomputer.