A Space-Time Multigrid Method for Space-Time Finite Element Discretizations of Parabolic and Hyperbolic PDEs

TL;DR

This paper introduces a space-time multigrid method for finite element discretizations of PDEs, demonstrating high scalability and efficiency on large-scale problems with complex coefficients using the deal.II library.

Contribution

It develops a novel space-time cell-wise additive Schwarz smoother and showcases its effectiveness in high-performance computing environments.

Findings

Achieves over a billion degrees of freedom per second.

Demonstrates high convergence rates on perturbed and heterogeneous meshes.

Effective for complex heat and wave equations in high-fidelity simulations.

Abstract

We present a space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order continuous and discontinuous variational time discretizations with spatial finite element discretizations. The effectiveness of multigrid methods in large-scale stationary problems is well established. However, their application in the space-time context poses significant challenges, mainly due to the construction of suitable smoothers. To address these challenges, we develop a space-time cell-wise additive Schwarz smoother and demonstrate its effectiveness on the heat and acoustic wave equations. The matrix-free framework of the {\ttfamily deal.II} library supports various multigrid strategies, including $h$-, $p$-, and $hp$-refinement across spatial and temporal dimensions. Extensive empirical evidence, provided through scaling and convergence tests on high-performance computing platforms, demonstrate high performance on perturbed meshes and problems with heterogeneous and discontinuous coefficients. Throughputs of over a billion degrees of freedom per second are achieved on problems with more than a trillion global degrees of freedom. The results prove that the space-time multigrid method can effectively solve complex problems in high-fidelity simulations and show great potential for use in coupled problems.