[2301.01753] Generalized Yee methods: Scalable symplectic finite element Maxwell solvers Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2301.01753 (math) [Submitted on 4 Jan 2023 ( v1 ), last revised 28 Apr 2026 (this version, v2)] Title: Generalized Yee methods: Scalable symplectic finite element Maxwell solvers Authors: Alexander S. Glasser , Hong Qin View a PDF of the paper titled Generalized Yee methods: Scalable symplectic finite element Maxwell solvers, by Alexander S. Glasser and Hong Qin View PDF HTML (experimental) Abstract: Yee's finite-difference method preserves two crucial properties of Maxwell's equations -- locality and symplecticity -- and thereby enjoys two computational advantages: scalability on high-performance architectures and long-time numerical accuracy. In this work, we show that Yee's method is a special case of a class of structure-preserving finite element methods -- termed generalized Yee methods (GYMs) -- that are designed to retain both crucial properties. GYMs are built from de Rham-conforming finite elements and achieve locality through sparse mass matrices and their sparse approximate inverses (SPAIs). We prove that the symplectic structure of GYMs is invariant under such sparse approximations, freeing the choice of sparsification strategy. We introduce a novel sparsification strategy, SPAI-OP, which concentrates accuracy at prescribed wave modes by operator probing. We further extend GYMs to structure-preserving electromagnetic particle-in-cell (PIC) methods, whose symplecticity over particle trajectories requires the smooth fields afforded by higher-order finite elements. GYMs therefore retain the computational virtues of Yee's method while enabling unstructured meshes, higher-order accuracy, spectral adaptivity, and symplectic particle coupling. Subjects: Numerical Analysis (math.NA) ; Computational Physics (physics.comp-ph) Cite as: arXiv:2301.01753 [math.NA] (or arXiv:2301.01753v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2301.01753 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexander S. Glasser [ view email ] [v1] Wed, 4 Jan 2023 18:35:51 UTC (2,153 KB) [v2] Tue, 28 Apr 2026 02:48:22 UTC (4,301 KB) Full-text links: Access Paper: View a PDF of the paper titled Generalized Yee methods: Scalable symplectic finite element Maxwell solvers, by Alexander S. Glasser and Hong Qin View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2023-01 Change to browse by: cs cs.NA math physics physics.comp-ph References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from