Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid (ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method. The performance of the algorithms are examined in solving the laminar flow over an air-foil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton-GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem. It is concluded that the Newton-GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied. (C) 2009 Elsevier Inc. All rights reserved.