Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods

We suggest a new multigrid preconditioning strategy for use in Jacobian-free Newton–Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretisations. To define the new preconditioner, use is made of an auxiliary first-order finite volume discretisation that refines the original DG mesh, but can still be implemented algebraically. As smoother, we consider the pseudo-time iteration W3 with a symmetric Gauss–Seidel-type approximation of the Jacobian. As a proof of concept numerical tests are presented for the one-dimensional Euler equations, demonstrating the potential of the new approach