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

Abstract

This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of diffusion. Gauss-Seidel relaxation converges ≈ 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

Efficient Relaxation Methods for High-Order Discretization of Steady Problems

“…However, just as for geometric multigrd methods the optimal mesh coarsening ratio is not always a priori clear, with multi−p methods the same question applies to the lower polynomial degree one ought to use. While some Fourier analysis for linear model equations has been carried out to assess convergence factors for multi−p methods [Atkins and Helenbrook (2005); Fidkowski et al (2005)], the issue of how many polynomial levels one ought to include for high m is still open in general, and likely problem and discretization dependent. We shall simply assume that in a two-grid cycle we use polynomial levels m and m c with 0 ≤ m c < m.…”

iHDG: An Iterative HDG Framework for Partial Differential Equations

“…Moreover the point or the block Gauss-Seidel scheme (for the trace system alone) requires a lot of communication between processors for calculations within an iteration. These aspects affect the scalabilty of these schemes to a large extent and in general are not favorable for parallelization [1,16].…”