A penalty-free bulk--surface CutFEM stabilized by lattice Green's function extensions

[2605.06329] A penalty-free bulk--surface CutFEM stabilized by lattice Green's function extensions Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2605.06329 (math) [Submitted on 7 May 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: A penalty-free bulk--surface CutFEM stabilized by lattice Green's function extensions Authors: Qing Xia View PDF HTML (experimental) Abstract: We introduce a penalty-free cut finite element method for surface elliptic problems coupled to a harmonic bulk field on a Cartesian grid. Instead of adding a stabilization term, the method restricts the active finite element space by a discrete bulk harmonic extension represented with the lattice Green's function, together with a local extrapolation near the interface. The resulting method is a symmetric Galerkin scheme posed on a reduced space embedded in the standard active-mesh finite element space. Under stated geometric, regularity, and approximation assumptions, we establish optimal $\mathcal{O}(h)$ and $\mathcal{O}(h^2)$ surface convergence rates, as well as robust, cut-independent algebraic conditioning. Furthermore, a density formulation based on lattice layer potentials is shown to act as an operator preconditioner; the single-layer parametrization yields an $\mathcal{O}(1)$ algebraically well-conditioned system without introducing any tunable parameters. Two-dimensional experiments on circular, deformed, and smooth nonconvex interfaces confirm the predicted convergence and robustness under changes in the cut position. Finally, a three-dimensional torus experiment demonstrates the identical construction using a seven-point bulk stencil and trilinear surface traces, exhibiting the expected optimal error rates. Subjects: Numerical Analysis (math.NA) ; Computational Physics (physics.comp-ph) MSC classes: 65N30, 65N12, 65F08, 35J05 Cite as: arXiv:2605.06329 [math.NA] (or arXiv:2605.06329v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2605.06329 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Qing Xia [ view email ] [v1] Thu, 7 May 2026 14:24:11 UTC (724 KB) [v2] Tue, 25 Aug 2026 08:31:12 UTC (1,825 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-05 Change to browse by: cs cs.NA math physics physics.comp-ph References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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