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‎references/index.html‎

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MargenbergEtAl2026
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<span id="MargenbergEtAl2026">N. Margenberg and M. Bause, “A Monolithic hp Space-Time Multigrid Preconditioned Newton-Krylov Solver for Space-Time FEM applied to the Incompressible Navier-Stokes Equations,” arXiv:2602.13841v1 [math.NA], 2026 [Online]. Available at: <a href="https://arxiv.org/abs/2602.13841v1" target="_blank">https://arxiv.org/abs/2602.13841v1</a></span>
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BibTeX entry <code>MargenbergEtAl2026</code>
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<pre class="abstract">@unpublished{MargenbergEtAl2026,
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author = {Margenberg, Nils and Bause, Markus},
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howpublished = {arXiv:2602.13841v1 [math.NA]},
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title = {A Monolithic hp Space-Time Multigrid Preconditioned Newton-Krylov Solver for Space-Time FEM applied to the Incompressible Navier-Stokes Equations},
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url = {https://arxiv.org/abs/2602.13841v1},
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year = {2026}
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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^\mathrmdisc 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.
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