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

TL;DR

This work develops a matrix-free space-time multigrid (STMG) framework for tensor-product space-time finite element discretizations of parabolic and hyperbolic PDEs, allowing high-order DG and CGP time discretizations to couple naturally with spatial FE. A space-time cell-wise additive Schwarz smoother serves as the key preconditioner within GMRES, with transfers and coarsening performed in space and time in a tensor-product fashion. Extensive HPC experiments on heat and acoustic wave problems, including highly heterogeneous coefficients and a structural health monitoring-inspired 3D scenario, show optimal convergence and strong scalability, achieving throughputs exceeding $10^9$ degrees of freedom per second on problems with more than $10^{12}$ global DOF. While higher-order discretizations yield better accuracy per work, the smoother cost becomes the main bottleneck, motivating future work on more efficient smoothers such as patch-based or $p$-multigrid strategies. Overall, the matrix-free STMG approach demonstrates robust, scalable performance and holds strong potential for large-scale coupled and multi-physics problems.

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.

Figures (10)

• Figure 1: A sketch of the STMG method, see also Algorithm \ref{['alg:stmg']}. The corrections are transferred by the prolongation operators and the residual is transferred by the restriction operators. On each level the error is smoothed by a single iteration of the additive Schwarz smoother. The preferred coarsening strategy, which is then used in Algorithm \ref{['alg:mg-sequence']}, is first in space and then in time.
• Figure 2: Calculated errors for $u$ for different orders of convergence on Cartesian meshes (top row) and perturbed meshes (bottom row) for $CG(k)-DG(k)$ discretizations of the heat equation. The expected orders of convergence $k+1$, represented by the triangles, match with the experimental orders.
• Figure 3: Calculated $L^2-L^2(u)$-errors of the heat equation for different polynomial orders plotted over the work \ref{['eq:work']} on Cartesian meshes (left) and perturbed meshes (right). The advantage of higher order discretizations can be observed.
• Figure 4: Calculated errors for the displacement $u$ for different orders on Cartesian (top row) and perturbed meshes (bottom row) for $CG(k)-DG(k)$ (dashed) and $CG(k)-CGP(k)$ (solid) discretizations of the wave equation. The expected orders of convergence $k+1$, match with the experimental orders.
• Figure 5: Calculated $L^2-L^2(u)$-errors of the wave equation for different orders plotted over the work on Cartesian meshes (left) and perturbed meshes (right). Higher-order discretizations are beneficial.