Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes

[2607.07963] Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2607.07963 (math) [Submitted on 8 Jul 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes Authors: Vitoriano Ruas View PDF HTML (experimental) Abstract: A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to its approximation by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed. These studies, due to authors such as Barrett and Elliott (1988), advocated the use of isoparametric finite elements with a single curvilinear edge or face fitting the true boundary at two or three vertexes and also at additional points on those curves or curved surfaces, so as to define non polynomial piecewise solutions with the optimal order of approximation. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting only of straight-edged elements, upon which the solution is a polynomial. Lost optimal orders of convergence, by virtue of the domain's approximation by a polytope, are recovered by means of the addition of terms to the bilinear form. These account for natural boundary conditions of the same type as the true ones, though to be prescribed on the approximating boundary instead. The new technique is applied to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results. Comments: This modified version was finalized mostly following suggestions and advice by a few experts in its field. Accordingly, the article's mathematical and computational contents changed to a large extent, as much as its title Subjects: Numerical Analysis (math.NA) MSC classes: 65N30, 74S05, 76M10, 78M10, 80M10 Cite as: arXiv:2607.07963 [math.NA] (or arXiv:2607.07963v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2607.07963 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Vittoriano Ruas [ view email ] [v1] Wed, 8 Jul 2026 22:34:29 UTC (60 KB) [v2] Fri, 28 Aug 2026 08:28:58 UTC (75 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-07 Change to browse by: cs cs.NA math 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)

Related documents

• Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes#682753
• Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes#609646
• Stream Computing Research and Practical Cartesian Meshing for Complex Urban Environments#493187
• Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes#453506
• Optimization of a Triangular Delaunay Mesh Generator using Reinforcement Learning#798589