[2608.14484] Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2608.14484 (math) [Submitted on 14 Aug 2026] Title: Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes Authors: Yuanhui Lin , Tao Lin , Xu Zhang , Minfu Feng View a PDF of the paper titled Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes, by Yuanhui Lin and Tao Lin and Xu Zhang and Minfu Feng View PDF HTML (experimental) Abstract: We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and $L^2$ norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes. Subjects: Numerical Analysis (math.NA) Cite as: arXiv:2608.14484 [math.NA] (or arXiv:2608.14484v1 [math.NA] for this version) https://doi.org/10.48550/arXiv.2608.14484 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xu Zhang [ view email ] [v1] Fri, 14 Aug 2026 17:00:15 UTC (886 KB) Full-text links: Access Paper: View a PDF of the paper titled Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes, by Yuanhui Lin and Tao Lin and Xu Zhang and Minfu Feng View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-08 Change to browse by: cs cs.NA math 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