Implicit Solvers

An elegant yet practical framework for the application of p- and h-Multigrid for DGFEM is first presented and analysed theoretically. After that a hybrid implicit-explicit p-Multigrid iteration strategy for the discretisation of the steady Euler equations with the discontinuous Galerkin finite element method is presented and investigated experimentally. The implicit strategy consists of an inexact damped Newton iteration, using an ILU(0)-preconditioned matrix-free GMRES method for the solution of the linear system. The size of the ILU preconditioner grows very fast with interpolation order. As a result the method becomes impractical already for moderate orders of interpolation. Therefore it is embedded in a FAS p-Multigrid iteration scheme. In this framework, the implicit solver is used only on the lowest order interpolation space, while on the higher order levels a Runge-Kutta local timestepping method is used. The fast convergence of the solution on the lowest order levels speeds up the convergence of the mainly explicit multilevel iterations considerably with respect to a purely explicit p-Multigrid solver.