Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems

TL;DR

This paper introduces a novel space-time parallel multigrid method for parabolic equations, utilizing high-order discontinuous Galerkin discretizations and a block Jacobi smoother, with proven convergence and scalable parallel implementation.

Contribution

It presents a new multigrid algorithm with detailed convergence analysis, optimal smoothing parameters, and demonstrates excellent scalability for parabolic problems.

Findings

Convergence analysis for heat equation

Optimal smoothing parameters identified

Excellent scalability demonstrated

Abstract

We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time, and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation, and determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel, and show with numerical experiments its excellent strong and weak scalability properties.