Analysis of p-multigrid solution schemes for discontinuous Galerkin discretizations of flow problems.

摘要

p-multigrid is a 'multigrid-like' algorithm used to obtain solutions to high-order hp-finite element discretizations. In this method convergence is accelerated by using coarse levels constructed by reducing the order, p, of the approximating polynomial. We have investigated p-multigrid coupled with preconditioned block relaxation schemes to obtain the steady-state solution to discontinuous Galerkin (DG) discretizations of the Euler equations. Block-diagonal, -line, and sweeping preconditioners, and also the alternate direction implicit (ADI), and the incomplete lower-upper (ILU(0)) preconditioners are considered. Relaxation schemes that approximately-invert (AI) the steady-state stiffness matrix and implicit psuedo time-advancing (ITA) schemes are Fourier analyzed and compared. In general, for orders of approximating polynomial p ≥ 2, the AI schemes perform better than the similarly preconditioned ITA schemes. The results show that p-multigrid iterations of the AI-ILU(0) scheme with under-relaxation o = 1/2 converge fastest and are the most robust of the schemes studied. Similar to prior observations by Helenbrook and Atkins p-multigrid was observed to behave anomalously when p transitions from 1 to 0. Using ideas from Helenbrook and Atkins correction for diffusion, and the streamwise upwind Petrov-Galerkin (SUPG) formulation, this anomalous behavior is corrected for the 1D convection equation. The correction is then extended to the 1D convection-diffusion equation.