{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-hp-multigrid-approach-for-space-time","title":"An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations","arxiv_id":"2502.09159","date":"2025-02-13","proceeding":null,"authors":["Nils Margenberg","Markus Bause","Peter Munch"],"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.","url_abs":"https://arxiv.org/abs/2502.09159v2","url_pdf":"https://arxiv.org/pdf/2502.09159v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-hp-multigrid-approach-for-space-time","repo_url":"https://github.com/nlsmrg/dealii-stfem","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}