Analysis of a hybrid p-Multigrid method for the discontinuous Galerkin discretisation of the Euler equations

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A hybrid implicit and explicit p-Multigrid iteration strategy for the discretisation of the steady Euler equations with the discontinuous Galerkin finite element method is presented. The implicit strategy consists of an inexact damped Newton iteration, using an ILU(0)-preconditioned GMRES method for the solution of the linear system. Since the size of the preconditioner grows quadratically (2D) / cubically (3D) with the interpolation order, this method becomes impractical for moderate to high order interpolations. Therefore the implicit method is embedded in a FAS multilevel iteration scheme, which is based on successive interpolation spaces. The implicit solver is only on the lowest order interpolation spaces, while on the higher order levels a cheap explicit method is used. The fast convergence of the solution on the non-coarsened levels speeds up the convergence of the multilevel iterations considerably. The interlevel transfer operations are based on L2 projections for the solutions and on a reformulation of the variational problem itself for the residual. These operators are straightforward to implement and localized per element. The convergence of the pMultigrid two-cycle algorithm using those operators is investigated theoretically and experimentally.