Application of p-multigrid to discontinuous Galerkin formulations of the Poisson equation

The p-multigrid is investigated as a method for solving discontinuous Galerkin (DG) formulations of the Poisson equation. Different relaxation schemes and basis sets are combined with the DG formulations to find the best performance. The damping factors of the schemes are determined using Fourier analysis for both one- and two-dimensional problems. One important finding is that the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability, these 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 outperforms 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.