An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations

TL;DR

This work develops an hp space-time multigrid method for tensor-product space-time finite element discretizations of the nonstationary Stokes equations. By combining geometric and polynomial coarsening with a matrix-free, cell-based space-time Vanka smoother, the hp STMG preconditions GMRES to achieve scalable performance on HPC platforms, even for problems with trillions of degrees of freedom. The approach yields optimal or near-optimal h-robustness and competitive p-robustness, with substantial throughput gains and effective time-space coupling. While the smoothing step remains the main computational bottleneck, the method demonstrates strong potential for large-scale fluid dynamics simulations and can be extended to nonlinear regimes and related coupled problems.

Abstract

We present a monolithic $hp$ space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}$ of elements in space and a discontinuous Galerkin (DG$(k)$) discretization in time with piecewise polynomials of order $k$. The key novelty of this work is the application of $hp$ multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal$.$II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \( hp \) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second.

Figures (5)

• Figure 1: Sketch of the $hp$ STMG of \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 smoothened by application of the Vanka operator \ref{['vanka0']}. The coarsening strategy which is used in \ref{['alg:CombineHierarchies']}, is first in space and then in time in combination with polynomial coarsening before geometric coarsening (cf. \ref{['itm:p1']}, \ref{['itm:p2']}).
• Figure 1: Strong scaling test results for the STMG algorithm with varying numbers of smoothing steps. The left plot shows the time to solution over the number of MPI processes. The dashed gray lines indicate the optimal scaling. The right plot depicts the degrees of freedom (dofs) processed per second over the number of MPI processes.
• Figure 2: Time spent in different parts of the lid-driven cavity flow simulation, executed on $18432$ MPI processes. The simulations were conducted for $c=7$, $r\in\{3,\,4\}$, $n_{\text{sm}}\in\{1,\,2,\,4\}$.
• Figure D.1: Calculated errors of the velocity and pressure in various norms (velocity: $L^2$, $L^{\infty}$ in space-time and the $L^2$-norm of the divergence in space-time, pressure: $L^2$ in space-time) for different polynomial orders. The expected orders of convergence, represented by the triangles, match with the experimental orders.
• Figure E.1: Normalized Pressure difference $p_{\text{diff}}(t)$ over time.

Theorems & Definitions (7)

• Remark 2.1: Choice of the finite element spaces in \ref{['Def:VhQh']}
• Remark 2.2: Function spaces and their tensor product structure
• Remark 3.3: Global linear system of Problem \ref{['Prob:GloAlg']}
• Definition 4.1: Restriction and prolongation
• Remark 4.2: Construction of cell and vertex star patches
• Remark 4.3: Choice of the iterative solver for \ref{['Eq:LSMG']}
• Remark A.1