Application of the p-Multigrid Algorithm to Discontinuous Galerkin Formulations of the Compressible Euler Equation

Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection–diffusion equation

“…The two-level cycle iteration gives an upper bound for the convergence rate because the second level equations are solved directly rather than iteratively. To be consistent with our earlier analyses on the subject [5,6,10], the above V-cycle incorporates a single pre-smoothing step and no post-smoothing We have verified that adding additional relaxations (smoothing) does not change the qualitative conclusions.…”

“…With x ¼ 2=3, the damping factors for p ¼ 1 to 0 are essentially the same as those in Table 2. We observed a similar difficulty with the p ¼ 1 to 0 transition in our study of p-multigrid solution methods for DG discretizations of the 2D Euler equations [10]. This apparently anomalous behavior of the p ¼ 1 to p ¼ 0 transition is problematic because it consequently limits the convergence rate of any cycle that coarsens to p ¼ 0.…”

“…We have also observed a similar difficulty in applications of p-multigrid to DG discretizations of the 2-D Euler equations [10]. Helenbrook and Atkins [6] discovered the underlying reason for this degradation and provided a correction that works well for diffusive problems.…”

“…Based on the order of the coarse space polynomial there are different p-multigrid strategies. The most frequently used algorithms employ a coarse space polynomial order of either p c ¼ p=2 [5,6,10], p c ¼ p À 1 [14,15], or p c þ 1 ¼ ðp þ 1Þ=2 [16]. Helenbrook and Atkins [6] have compared these algorithms for the Poisson equation and their results show that p c ¼ p À 1 offers only a small improvement over p c ¼ p=2.…”

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

“…Most of this work is either with spectral element methods or with the discontinuous Galerkin method for fluid flow problems. At this point it becomes difficult to be exhaustive in the citations, but a sampling of the work can be found in [26][27][28][29][30][31][32][33][34][35][36][37].…”