Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

Abstract

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

“…The method provides mesh independent convergence and can be implemented with little storage or work overhead. This paper describes numerical experiments with P-multigrid that corroborate the analysis presented in the companion paper, 20 and validate the present implementation. This paper will also discuss attributes of the various combinations of relaxation schemes, discretization methods, and P-multigrid methods that become apparent when considering the implementation of these methods.…”

“…[5][6][7]12 Efforts to develop efficient implicit solvers for DG have included optimized relaxation schemes, 11 analysis of traditional geometric multigrid, 13 GMRES, [14][15][16] and P-multigrid. [17][18][19][20] Analysis of geometric multigrid indicates mesh that independent results are possible; however, the required restriction operators are complex and would be difficult to implement for general unstructured grids. A large effort in GMRES can be found in open literature; however the method incurs a large storage penalty, and none have achieved mesh independent convergence for high Reynolds number flows (or other highly stiff cases).…”

Super-Convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

“…10, a Fourier analysis of DG applied to diffusion indicated that all three discretizations described in the previous section have a super-convergence property. In that work, the analysis was performed for p = 4 only, and it was found that the DG-IP discretization converged at a rate of p = 9, while the LDG-C and LDG-OS discretizations converged at a rate of p = 12.…”

“…[4][5][6][7][8][9][10] In particular, a spatial eigenvalue analysis by Hu and Atkins 7,8 for scalar convection and acoustic propagation clearly shows that the amplitude and phase errors converge respectively at rates of 2p + 1 and 2p + 2. The analysis demonstrated the superconvergence property for a range of p, and it was conjectured that it held for all p. This conjecture was later proven in Ref.…”

“…In Ref. 10, a spatial Fourier analysis of several variations of DG applied to diffusion for p = 4 predicted the convergence of the temporal eigenvalue to be p = 9 or p = 12 depending on the form of the discretization. It has been conjectured that perhaps different DG discretizations converged at rates of 2p + 1 or 3p.…”