{"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/low-order-preconditioning-of-the-stokes","title":"Low-order preconditioning of the Stokes equations","arxiv_id":"2103.11967","date":"2021-03-22","proceeding":null,"authors":["Alexey Voronin","Yunhui He","Scott MacLachlan","Luke N. Olson","Ray Tuminaro"],"abstract":"A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor-Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the $\\boldsymbol{ \\mathbb{Q}}_1iso\\boldsymbol{ \\mathbb{Q}}_2/ \\mathbb{Q}_1$ discretization of the Stokes operator as a preconditioner for the $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ system, our ultimate motivation is to apply algebraic multigrid within solvers for $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ systems via the $\\boldsymbol{ \\mathbb{Q}}_1iso\\boldsymbol{ \\mathbb{Q}}_2/ \\mathbb{Q}_1$ discretization, which will be considered in a companion paper.","url_abs":"https://arxiv.org/abs/2103.11967v2","url_pdf":"https://arxiv.org/pdf/2103.11967v2.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":"low-order-preconditioning-of-the-stokes","repo_url":"https://github.com/lexeyV/Stokes_isoQ2Q1","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}