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

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.