An h-multigrid approach for high-order discountinuous Galerkin methods

The thesis presents the development of an h-multigrid solver for high-order accurate discontinuous Galerkin (DG) discretizations of non-linear systems of conservations laws on unstructured grids. For this purpose, a high-order DG discretization on polyhedral grids is developed first and it is applied to the compressible Navier-Stokes equations. The proposed method employs shape functions, consisting of complete polynomials defined in the real space, which are hierarchic and orthonormal on arbitrarily shaped elements. As regards the discretization of the viscous terms of the Navier-Stokes equations, we use the well-known method introduced in [6] with suitably enlarged values of the stability parameter reported in [1]. The accuracy and the convergence properties of the method have been tested against classical inviscid problems such as the transonic Ringleb flow and the subsonic flow over a Gaussian bump. The Helmholtz problem and the solution of the Navier-Stokes equations around a NACA0012 airfoil have been used as viscous tests. Then we present the development of an h-multigrid method where coarse grid levels are constructed by agglomerating neighbouring elements of the fine grid. First, an elegant yet practical set of transfer operators is derived for general space settings and for the current one. After that, a quasi-implicit multistage h-multigrid iteration strategy for the discontinuous Galerkin discretization of the steady Euler equations is developed and numerically investigated. Results are presented for a subsonic flow over a NACA0012 airfoil at 2o of incidence. These results highlight the main properties of the developed multigrid scheme and its different behaviour with respect to the classic p-multigrid scheme.