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A Monolithic hp Space-Time Multigrid Preconditioned Newton-Krylov Solver for Space-Time FEM applied to the Incompressible Navier-Stokes Equations

Nils Margenberg, Markus Bause

TL;DR

The paper addresses scalable, high-order simulation of incompressible flows by recasting the Navier–Stokes problem into a fully coupled space-time variational formulation and solving it with a matrix-free, monolithic $hp$ space-time multigrid preconditioned Newton–Krylov method. It introduces slab-wise tensor-product spaces with $ ext{DG}(k)$ time discretization and mapped inf-sup stable spatial pairs, and develops an $hp$ STMG preconditioner combining geometric and polynomial coarsening, a Vanka smoother, and a midpoint surrogate to keep the convection Jacobian affordable. The main contributions are the demonstration of $h$- and $p$-robust convergence and $ ext{Re}$-robust solver performance across a wide Reynolds range, plus strong parallel scalability and high throughput in large-scale MPI runs. The practical impact is a robust, scalable solver framework for nonlinear space-time FEM of NSE that leverages tensor-product structure and matrix-free evaluation to enable efficient HPC-based simulations with high-order accuracy.

Abstract

We present a monolithic hp space-time multigrid method (hp-STMG) for tensor-product space-time finite element discretizations of the incompressible Navier-Stokes equations. We employ mapped inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\mathrm{disc}}$ in space and a slabwise discontinuous Galerkin DG($k$) discretization in time. The resulting fully coupled nonlinear systems are solved by Newton-GMRES preconditioned with hp-STMG, combining geometric coarsening in space with polynomial coarsening in space and time. Our main contribution is an hp-robust and practically efficient extension of space-time multigrid to Navier-Stokes: matrix-free operator evaluation is retained via column-wise, state-dependent spatial kernels; the nonlinear convective term is handled by a reduced, order-preserving time quadrature. Robustness is ensured by an inexact space-time Vanka smoother based on patch models with single time point evaluation. The method is implemented in the matrix-free multigrid framework of deal.II and demonstrates h- and p-robust convergence with robust solver performance across a range of Reynolds numbers, as well as high throughput in large-scale MPI-parallel experiments.

A Monolithic hp Space-Time Multigrid Preconditioned Newton-Krylov Solver for Space-Time FEM applied to the Incompressible Navier-Stokes Equations

TL;DR

The paper addresses scalable, high-order simulation of incompressible flows by recasting the Navier–Stokes problem into a fully coupled space-time variational formulation and solving it with a matrix-free, monolithic space-time multigrid preconditioned Newton–Krylov method. It introduces slab-wise tensor-product spaces with time discretization and mapped inf-sup stable spatial pairs, and develops an STMG preconditioner combining geometric and polynomial coarsening, a Vanka smoother, and a midpoint surrogate to keep the convection Jacobian affordable. The main contributions are the demonstration of - and -robust convergence and -robust solver performance across a wide Reynolds range, plus strong parallel scalability and high throughput in large-scale MPI runs. The practical impact is a robust, scalable solver framework for nonlinear space-time FEM of NSE that leverages tensor-product structure and matrix-free evaluation to enable efficient HPC-based simulations with high-order accuracy.

Abstract

We present a monolithic hp space-time multigrid method (hp-STMG) for tensor-product space-time finite element discretizations of the incompressible Navier-Stokes equations. We employ mapped inf-sup stable pairs in space and a slabwise discontinuous Galerkin DG() discretization in time. The resulting fully coupled nonlinear systems are solved by Newton-GMRES preconditioned with hp-STMG, combining geometric coarsening in space with polynomial coarsening in space and time. Our main contribution is an hp-robust and practically efficient extension of space-time multigrid to Navier-Stokes: matrix-free operator evaluation is retained via column-wise, state-dependent spatial kernels; the nonlinear convective term is handled by a reduced, order-preserving time quadrature. Robustness is ensured by an inexact space-time Vanka smoother based on patch models with single time point evaluation. The method is implemented in the matrix-free multigrid framework of deal.II and demonstrates h- and p-robust convergence with robust solver performance across a range of Reynolds numbers, as well as high throughput in large-scale MPI-parallel experiments.
Paper Structure (33 sections, 2 theorems, 74 equations, 3 figures, 14 tables)

This paper contains 33 sections, 2 theorems, 74 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Continuous and discrete problem
  3. Continuous problem
  4. Space-time finite element discretization
  5. Algebraic system
  6. Preliminaries
  7. Assembly of the bilinear and linear terms in \ref{['eq:DNSE']}
  8. Assembly of the nonlinear convective terms in \ref{['eq:DNSE']}
  9. Temporal quadrature approximation of the nonlinear term
  10. Algebraic form of the discrete problem
  11. Solution of the nonlinear system of equations
  12. Inexact Newton--Krylov with Armijo globalization
  13. Preconditioning of GMRES by $hp$ space-time multigrid (STMG)
  14. Multilevel hierarchies and discrete spaces
  15. Grid transfer operators
  16. ...and 18 more sections

Key Result

Lemma 3.2

For $k\in \mathbb{N}_0$, suppose that $\boldsymbol u_{\tau h} \in Y_\tau^k(I) \otimes \boldsymbol V_h^{\operatorname{div}}$ satisfies, for $l=0,\ldots,k$, where $\widetilde{\Delta}_h$ denotes the discrete Stokes operator (cf. HeywoodRannacher1982), and $A_k$ is independent of the mesh sizes $\tau_n$ and $h$. For $\boldsymbol w_{\tau h} \in Y_\tau^k(I) \otimes \boldsymbol V_h^{r+1}$ let For the $

Figures (3)

  • Figure 1: Strong scaling test results for the STMG algorithm with varying numbers of smoothing steps (anchored to the $c=7$ case at $18432$ MPI ranks, cf. Table \ref{['tab:lid-throughput-stokes']}). 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: Relative time spent in dominant parts of the Navier-Stokes solve on $18432$ MPI ranks. The cell-wise Vanka smoother is dominated by the expensive rebuild and application of the Vanka smoother. Data shown for $c=7$, $r\in\{2,\,3,\,4\}$, $n_{\text{sm}}\in\{1,\,2\}$.
  • Figure C.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.

Theorems & Definitions (12)

  • Remark 2.1: Non mixed-type boundary condition
  • Remark 2.2: Tensor products and Bochner spaces
  • Remark 2.4: Preservation of tensor-product structure
  • Remark 3.1
  • Lemma 3.2: Gauss--Radau quadrature error for divergence-form of convection
  • Proof 1
  • Remark 3.3: On the result of Lemma \ref{['lem:GR-div']}
  • Definition 3.5: Global algebraic Navier--Stokes problem
  • Remark 4.1
  • Lemma 4.2: Characterization of local Vanka surrogate
  • ...and 2 more