Multigrid preconditioning for a space-time spectral-element discontinuous-Galerkin solver

In this work we examine a multigrid preconditioning approach in the context of a highorder tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scaleresolving solutions of the Navier–Stokes equations: separated low-Mach unsteady flow over an airfoil from laminar to turbulent flow. A reduction in the number of fine space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined.