{"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/localized-orthogonal-decomposition-methods-vs","title":"Localized Orthogonal Decomposition Methods vs. Classical FEM for the Gross-Pitaevskii Equation","arxiv_id":"2403.11268","date":"2024-03-17","proceeding":null,"authors":["Christian Döding"],"abstract":"The time-dependent Gross-Pitaevksii equation (GPE) is a nonlinear Schr\\\"odinger equation which is used in quantum physics to model the dynamics of Bose-Einstein condensates. In this work we consider numerical approximations of the GPE based on a multiscale approach known as the localized orthogonal decomposition. Combined with an energy preserving time integrator one derives a method which is of high order in space and time under mild regularity assumptions. In previous work, the method has been shown to be numerically very efficient compared to first order Lagrange FEM. In this paper, we further investigate the performance of the method and compare it with higher order Lagrange FEM. For rough problems we observe that the novel method performs very efficient and retains its high order, while the classical methods can only compete well for smooth problems.","url_abs":"https://arxiv.org/abs/2403.11268v2","url_pdf":"https://arxiv.org/pdf/2403.11268v2.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":"localized-orthogonal-decomposition-methods-vs","repo_url":"https://github.com/cdoeding/lodvsfem","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}