hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I: Multilevel analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space–time discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge–Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space–time discontinuous Galerkin discretization of the advection–diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for steady state and time-dependent problems, and low and high cell Reynolds numbers, including highly stretched meshes.