Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

TL;DR

This paper introduces a robust multigrid method with a Schwarz smoother for 3D Cartesian DG formulations of the Poisson equation, achieving fast convergence and scalability for high polynomial degrees and anisotropic grids.

Contribution

It develops a polynomial multigrid approach with a weighted Schwarz smoother for 3D interior penalty DG discretizations, demonstrating robustness and efficiency up to high polynomial degrees and anisotropic meshes.

Findings

Achieves residual reductions of about two orders of magnitude within one V-cycle.

Robust with respect to mesh size and polynomial degree up to P=32.

Linear runtime scaling observed in numerical experiments.

Abstract

We present a polynomial multigrid method for the nodal interior penalty formulation of the Poisson equation on three-dimensional Cartesian grids. Its key ingredient is a weighted overlapping Schwarz smoother operating on element-centered subdomains. The MG method reaches superior convergence rates corresponding to residual reductions of about two orders of magnitude within a single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz order, at least up to ${P=32}$. Rigorous exploitation of tensor-product factorization yields a computational complexity of $O(PN)$ for $N$ unknowns, whereas numerical experiments indicate even linear runtime scaling. Moreover, by allowing adjustable subdomain overlaps and adding Krylov acceleration, the method proved feasible for anisotropic grids with element aspect ratios up to 48.