Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

TL;DR

This paper introduces a polynomial multigrid method for high-order discontinuous Galerkin discretizations of the Poisson equation, achieving robust and efficient convergence on Cartesian grids with high aspect ratios.

Contribution

It develops a multigrid solver with novel Schwarz smoothers that are robust for high polynomial degrees and aspect ratios, with proven optimal complexity.

Findings

Achieves convergence rates robust to mesh size and polynomial degree

Computational complexity scales linearly with unknowns and polynomial degree

Suitable for grids with high aspect ratios up to 32

Abstract

We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to ${P=32}$. Consequent structure exploitation yields a computational complexity of $O(PN)$, where $N$ is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32.