p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

We present a p-multigrid solution algorithm for a high-order discontinuous Galerkin finite element discretization of the compressible Navier–Stokes equations. The algorithm employs an element line Jacobi smoother in which lines of elements are formed using coupling based on a p = 0 discretization of the scalar convection–diffusion equation. Fourier analysis of the two-level p-multigrid algorithm for convection–diffusion shows that element line Jacobi presents a significant improvement over element Jacobi especially for high Reynolds number flows and stretched grids. Results from inviscid and viscous test cases demonstrate optimal hp + 1 order of accuracy as well as p-independent multigrid convergence rates, at least up to p = 3. In addition, for the smooth problems considered, p-refinement outperforms h-refinement in terms of the time required to reach a desired high accuracy level.