[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85933-en":3,"doc-seo-85933-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85933,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Comparing Algebraic Cubature Rules on Spline Curved Elements","The paper compares four constructions of algebraic cubature rules for planar curved elements whose boundaries are tracked by spline curves. The approaches are built on Green’s theorem combined with polynomial approximation tools: Gaussian quadrature, Tchakaloff moment matching via nonnegative least squares, discretized Chebyshev hyperinterpolation, and Fekete-like interpolation using QR with column pivoting. Each method’s stability, node location behavior (internal vs. external), cardinality, and construction cost are analyzed for convex and concave curved polytopal element applications. Open-source Matlab codes are provided.","arXiv :2607 . 10485v1 [math .NA] 11 Jul 2026  \nComparing algebraic cubature rules on spline curved elements  \nAlvise Sommarivaa , Marco Vianelloa  \na Dept. of Mathematics, University of Padova, via Trieste 63, 35121 Padova (Italy)  \nAbstract  \nWe compare four methods for the construction of algebraic cubature rules on planar elements, whose boundary is tracked by splines. The methods, that we have developed over the last two decades, are based on Green theorem together with some cornerstones of polynomial approximation theory: Gaussian quadrature, Tchakaloff theorem, discretized Chebyshev expansion (hyperinterpolation), Fekete-like interpolation. We discuss their advantages and drawbacks in view of the application to curved polytopal element methods. We have also made freely available at a single site the corresponding open-source Matlab codes.  \nKeywords: Algebraic cubature rules, curved polytopal element methods, spline curved elements.  \n1. Introduction  \nThe availability of efficient cubature methods is a relevant issue within FEM and VEM methods on nonstandard elements, especially for curved polytopal elements in high-order instances. Indeed, computation of stiffness and mass matrices can still represent a bottleneck, that has to be coped with by specialized cubature rules. During the last two decades, a devoted literature has been growing on this topic, with some attention on avoiding subtriangulations or in general subtessellations; with no pretence of exhaustivity, we may quote [1, 3, 4, 9, 10, 11, 15, 16, 17, 22, 24, 27, 30, 36, 39] and the references therein.  \nIn this respect, the main purpose of this paper is to give a guided tour in the use of some specific codes, that we developed over the last twenty years, for algebraic cubature on a general curved element E ⊂ R2 (either convex or concave), in the case when its piecewise regular Jordan boundary ∂E is tracked by splines. Such codes produce weights {ws } and nodes {Ps = (xs , ys)} such that  \nM  \nZE f (x, y) dxdy = Xs=1 ws f (Ps) ∀f ∈ Pn , (1)  \n(where Pn denotes the space of bivariate polynomials with total degree not exceeding n), and are here named correspondingly to the construction approach:  \n• Spline-Gauss (2007-2009) [35, 36]: collection of product-like Gaussian rules by Green theorem and numerical primitives (possible negative weights and external nodes on concave elements, high cardinality, empirical stability in general and theoretical stability on convex elements, low construction cost) .  \n• Spline-Tchakaloff (2021 & 2024) [39, 40]: moment-matching by Green theorem and NonNegative Least Squares, supported at Tchakaloff sets (subtessellation-free, positive weights and internal nodes, low cardinality, theoretical stability, high construction cost) .  \n∗ Corresponding author: Marco Vianello  \nEmail addresses: [alvise@math.unipd.it](alvise@math.unipd.it) (Alvise Sommariva), [marcov@math.unipd.it](marcov@math.unipd.it) (Marco Vianello)  \n• Spline-Chebyshev (2025-2026) [27, 28, 41]: cheap moment-matching by Green theorem and discretized Chebyshev orthogonal expansion (hyperinterpolation), supported at a minimal cubature set for the Chebyshev measure in a bounding box (subtessellation-free, some external nodes and possible negative weights, low cardinality, theoretical stability, low construction cost) .  \n• Spline-Fekete (2009 & 2026) [37]: moment-matching by Green theorem and QR factorization with column pivoting, supported at approximate Fekete sets (subtessellation-free, possible negative weights but internal nodes, low cardinality, empirical stability, moderate construction cost) .  \nBy stability of the cubature rule we mean the fact that kw k 1 is bounded with n, and more precisely that, also in the presence of some negative weights, the stability parameter  \nσ = PP~~s~~s|~~ ~~wsws| = Par~~s~~ea|~~ ~~w(sE ≥ 1 (2)  \nstays relatively close to the optimal value 1 (positive weights), at least for the degrees of interest in FEMand VEM applications.  \nIn the seque","cbCaiaUeiiEXwTsI","https://ap.wps.com/l/cbCaiaUeiiEXwTsI","pdf",1224798,1,14,"English","en",105,"# Introduction\n# The four cubature rules","[{\"question\":\"What problem does the paper address in FEM/VEM on curved polytopal elements?\",\"answer\":\"It targets the challenge that computing stiffness and mass matrix entries can be bottlenecked, motivating efficient cubature rules on curved elements with spline-tracked boundaries.\"},{\"question\":\"Which four cubature-rule constructions are compared?\",\"answer\":\"The paper compares Spline-Gauss, Spline-Tchakaloff, Spline-Chebyshev, and Spline-Fekete, each based on Green’s theorem plus different polynomial-approximation foundations.\"},{\"question\":\"How is polynomial integration reduced when the element boundary is described by splines?\",\"answer\":\"Restricting a bivariate polynomial to each spline arc turns the integrand into a univariate polynomial in the spline parameter, enabling boundary integration through spline-arc quadrature with appropriate nodes and 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problem does the paper address in FEM/VEM on curved polytopal elements?","Question",{"text":75,"@type":76},"It targets the challenge that computing stiffness and mass matrix entries can be bottlenecked, motivating efficient cubature rules on curved elements with spline-tracked boundaries.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which four cubature-rule constructions are compared?",{"text":80,"@type":76},"The paper compares Spline-Gauss, Spline-Tchakaloff, Spline-Chebyshev, and Spline-Fekete, each based on Green’s theorem plus different polynomial-approximation foundations.",{"name":82,"@type":73,"acceptedAnswer":83},"How is polynomial integration reduced when the element boundary is described by splines?",{"text":84,"@type":76},"Restricting a bivariate polynomial to each spline arc turns the integrand into a univariate polynomial in the spline parameter, enabling boundary integration through spline-arc quadrature with appropriate nodes and 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