{"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/projection-methods-in-the-context-of-nematic","title":"Projection Methods in the Context of Nematic Crystal Flow","arxiv_id":"2502.08571","date":"2025-02-12","proceeding":null,"authors":["Maximilian E. V. Reiter"],"abstract":"We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including a non-convex unit-sphere constraint. As predictor step we propose a linear semi-implicit Finite Element discretization which naturally offers a local orthogonality relation between the approximate director field and its time derivative. Afterwards an explicit discrete projection onto the unit-sphere constraint is applied without increasing the modeled energy. For the Finite Element approximation of the director field, we compare the usage of a discrete inner product, usually referred to as mass-lumping, for a globally continuous, piecewise linear discretization to a piecewise constant, discontinuous Galerkin approach. Discrete well-posedness results and energy laws are established. Conditional convergence of the approximate solutions to energy-variational solutions of the Ericksen-Leslie equations is shown for a time-step restriction, see Theorems 1 and 2. Computational studies indicate the efficiency of the proposed linearization and the improved accuracy by including a projection step in the algorithm.","url_abs":"https://arxiv.org/abs/2502.08571v1","url_pdf":"https://arxiv.org/pdf/2502.08571v1.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":"projection-methods-in-the-context-of-nematic","repo_url":"https://github.com/Max-Reiter-math/ericksen_leslie_x","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}