In this talk, we focus on a p-multigrid solution algorithm for high-order discontinuous Galerkin (DG) finite element discretizations of the Euler and compressible Navier-Stokes equations. p-Multigrid, or multi-order, solution strategies have been studied by other authors, including Helenbrook, Mavriplis, and most recently Bassi and Rebay. Common features of this method for high-order DG include ease of implementation and order-independent convergence rates. A key aspect of our p-multigrid implementation is the use of an element-line Jacobi smoother instead of the standard element-Jacobi. The element-line Jacobi smoother consists of solving implicitly on lines of elements formed using coupling based on a p = 0 discretization of the scalar convection-diffusion equation. This choice of elemental coupling allows for the removal of stiffness associated with strong convection or regions of high grid anisotropy frequently required in viscous layers. A line creation algorithm is presented for general unstructured meshes, showing how unique lines can be obtained in two and three dimensions for a given elemental coupling.

Using Fourier analysis for scalar convection-diffusion, we demonstrate that the higher-order DG discretizations can be stably marched for all orders with element Jacobi and element-line Jacobi schemes without the use of multi-stage iterations. p-Multigrid is then applied with the element-line smoother to inviscid and viscous test cases, in two and three dimensions. Results demonstrate optimal order of accuracy of the discretization, as well as p-independent multigrid convergence rates. h-dependence is observed, although it is not found to be strong for many practical problems. Finally, for the smooth problems considered, p-refinement outperforms h-refinement in terms of the time required to reach a desired high accuracy level.