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arxiv: 2411.03607 · v1 · pith:YTW3I3PJnew · submitted 2024-11-06 · 🧮 math.NA · cs.NA

Upper bound of high-order derivatives for Wachspress coordinates on polytopes

Pengjie Tian , Yanqiu Wang This is my paper

Pith reviewed 2026-05-25 08:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Wachspress coordinatesgeneralized barycentric coordinateshigh-order derivativesupper boundspolytopal finite elementsshape-regularityconvex polytopesfourth-order elliptic equations
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The pith

Upper bounds are derived for high-order derivatives of Wachspress generalized barycentric coordinates on simple convex polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes upper bounds on derivatives of any order for Wachspress generalized barycentric coordinates on simple convex polytopes in dimension d at least 1. These bounds fill a gap needed for H^k error estimates when k exceeds 1, allowing convergence analysis for higher-order problems such as fourth-order elliptic equations. The derivation relies on the explicit rational form of the coordinates together with geometric properties of the polytopes. A secondary contribution compares and relates several shape-regularity conditions for such polytopes using tools from convex geometry.

Core claim

For Wachspress generalized barycentric coordinates on simple convex d-dimensional polytopes that satisfy one of the shape-regularity conditions, the supremum norm of any k-th order derivative is bounded by a constant that depends only on d, k, and the shape-regularity parameter, and is independent of the particular polytope in the admissible class.

What carries the argument

Upper bounds on high-order derivatives of Wachspress generalized barycentric coordinates, obtained from their explicit formula combined with the geometry of simple convex polytopes under shape-regularity.

Load-bearing premise

The polytopes must be simple and convex and must satisfy one of the shape-regularity conditions for the uniform derivative bounds to hold.

What would settle it

A sequence of shape-regular simple convex polytopes on which the k-th derivative of some Wachspress coordinate grows without bound as the diameter tends to zero would disprove the claimed upper bound.

Figures

Figures reproduced from arXiv: 2411.03607 by Pengjie Tian, Yanqiu Wang.

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read the original abstract

The gradient bounds of generalized barycentric coordinates play an essential role in the $H^1$ norm approximation error estimate of generalized barycentric interpolations. Similarly, the $H^k$ norm, $k>1$, estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex $d$-dimensional polytopes, $d\ge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper derives explicit upper bounds on the high-order derivatives of Wachspress generalized barycentric coordinates for simple convex d-dimensional polytopes (d ≥ 1) under stated shape-regularity conditions. It also compares and clarifies relations among various shape-regularity notions for such polytopes using tools from convex geometry. The derivative bounds are positioned to support optimal convergence analysis for Wachspress-based polytopal finite-element approximations of fourth-order elliptic problems.

Significance. If the stated bounds hold under the geometric hypotheses, the work supplies a missing ingredient for H^k (k > 1) error estimates of generalized barycentric interpolants, directly enabling optimal-rate analysis of fourth-order problems on polytopal meshes. The secondary clarification of shape-regularity conditions via convex geometry is a useful service to the literature on polytopal FEM.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise form of the derivative bound (e.g., the dependence on dimension d, the order k, and the shape-regularity parameter) rather than only describing its existence.
  2. Notation for the shape-regularity constants (e.g., the various equivalent conditions) should be introduced once in a dedicated subsection and then used consistently; cross-references to the comparison results would improve readability.
  3. The manuscript would benefit from a short remark on whether the derived constants are sharp or merely sufficient, even if a sharpness proof lies outside the present scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The assessment that the derivative bounds supply a missing ingredient for H^k error estimates is encouraging, as is the recognition of the shape-regularity clarification as a service to the polytopal FEM literature.

Circularity Check

0 steps flagged

No significant circularity; derivation from convex geometry

full rationale

The paper presents a derivation of explicit upper bounds on high-order derivatives of Wachspress generalized barycentric coordinates for simple convex polytopes under stated shape-regularity conditions from convex geometry. The abstract frames the result as a proof usable for finite element error estimates, with a secondary clarification of geometric conditions via convex geometry knowledge. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the central claim remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The work relies on standard results from convex geometry for the shape-regularity comparison, but these are not enumerated.

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Betke and M

    U. Betke and M. Henk, A generalization of Steinhagen’s theorem, Abh. Math. Sem. Univ. Hamburg, 63:165-176, 1993

    work page 1993

  2. [2]

    Brenner and L.R

    S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2007

    work page 2007

  3. [3]

    Brøndsted, An Introduction to Convex Polytopes, Springer, New York, 1983

    A. Brøndsted, An Introduction to Convex Polytopes, Springer, New York, 1983. 22 10-3 10-2 h* 102 103 104 = (1,0) , (0,1) 10-3 10-2 h* 104 106 108 = (2,0),(1,1),(0,2) 10-3 10-2 h* 105 109 1013 = (3,0),(2,1),(1,2),(0,3) Fig. 5.12 . From the first to the last panel, relation between Λα and h∗ with |α| = 1, 2, 3 on K3

    work page 1983

  4. [4]

    Chen and Y

    W. Chen and Y. Wang, H1, H(curl) and H(div) conforming elements on polygon-based prisms and cones, Numer. Math., 145:973-1004, 2020

    work page 2020

  5. [5]

    Chen and Y

    X. Chen and Y. Wang, A conforming quadratic polygonal element and its application to stokes equations, J. Comput. Math., 40(4):624-648, 2022

    work page 2022

  6. [6]

    Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

    P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

    work page 1978

  7. [7]

    Constantine and T.H

    G.M. Constantine and T.H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc., 348:503-520, 1996

    work page 1996

  8. [8]

    Danzer, B

    L. Danzer, B. Gr¨ unbaum and V. Klee, Helly’s theorem and its relatives, In: AMS Symposium on Convexity, Seattle, Proceedings of Symposium on Pure Mathematics, Amer. Math. Soc., Providence, RI, 7:101-180, 1963

    work page 1963

  9. [9]

    Q. Du, V. Faber and M. Gunzburger, Centroidal Voronoi tessellations: applications and algo- rithms, SIAM Rev., 41(4):637-676, 1999

    work page 1999

  10. [10]

    Eggleston, Convexity, Cambridge University Press, Cambridge, 1958

    H.G. Eggleston, Convexity, Cambridge University Press, Cambridge, 1958

    work page 1958

  11. [11]

    Floater, Mean value coordinates, Comput

    M.S. Floater, Mean value coordinates, Comput. Aided Geom. Design, 20:19-27, 2003

    work page 2003

  12. [12]

    Floater, Wachspress and mean value coordinates, in Approximation Theory XIV: San Antonio, Springer, 81-102, 2013

    M.S. Floater, Wachspress and mean value coordinates, in Approximation Theory XIV: San Antonio, Springer, 81-102, 2013

    work page 2013

  13. [13]

    Floater, Generalized barycentric coordinates and applications, Acta

    M.S. Floater, Generalized barycentric coordinates and applications, Acta. Numer., 24:161-214, 2015

    work page 2015

  14. [14]

    Floater, A

    M.S. Floater, A. Gillette and N. Sukumar, Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numer. Anal., 52(1):515-532, 2014

    work page 2014

  15. [15]

    Floater, K

    M.S. Floater, K. Hormann and G. K´ os, A general construction of barycentric coordinates over convex polygons, Adv. Comput. Math., 24:311-331, 2006

    work page 2006

  16. [16]

    Floater and M

    M.S. Floater and M. Lai, Polygonal spline spaces and the numerical solution of the poisson equation, SIAM J. Numer. Anal., 54(2):797-824, 2016

    work page 2016

  17. [17]

    Gillette and C

    A. Gillette and C. Bajaj, A generalization for stable mixed finite elements, Proc. 14th ACM Symp. Solid Phys. Modeling., 41-50, 2010

    work page 2010

  18. [18]

    Gillette and C

    A. Gillette and C. Bajaj, Dual formulations of mixed finite element methods with applications, Comput. Aided Des., 43(10):1213-1221, 2011

    work page 2011

  19. [19]

    Gillette, A

    A. Gillette, A. Rand and C. Bajaj, Error estimates for generalized barycentric interpolation, Adv. Comput. Math., 37:417-439, 2012. 23 10-3 10-2 h* 101 104 = (1,0) 10-3 10-2 h* 102 105 108 = (2,0) 10-3 10-2 h* 102 105 109 1012 = (3,0) Fig. 5.13 . Relation between maxx∈K |Dαϕi(x)| and h∗ for α = (1, 0), (2, 0), (3, 0) on K3. 10-4 10-3 10-2 h* 101 102 103 1...

    work page 2012

  20. [20]

    Gordon and J.A

    W.J. Gordon and J.A. Wixom, Pseudo-harmonic interpolation on convex domains, SIAM J. Numer. Anal., 11(5):909-933, 1974

    work page 1974

  21. [21]

    Gr¨ unbaum, Convex Polytopes (2nd ed.), Springer, New York, 2003

    B. Gr¨ unbaum, Convex Polytopes (2nd ed.), Springer, New York, 2003

    work page 2003

  22. [22]

    Hormann and N

    K. Hormann and N. Sukumar (Eds.), Generalized Barycentric Coordinates in Computer Graph- ics and Computational Mechanics (1st ed.), CRC press, Boca Raton, 2017

    work page 2017

  23. [23]

    Huang and Y

    W. Huang and Y. Wang, Anisotropic mesh quality measures and adaptation for polygonal meshes, J. Comput. Phys., 410:109368, 2020

    work page 2020

  24. [24]

    Martin, P

    S. Martin, P. Kaufmann, M. Botsch, M. Wicke and M. Gross, Polyhedral finite elements using harmonic basis functions, Comput. Graph. Forum, 27:1521-1529, 2008

    work page 2008

  25. [25]

    Milbradt and T

    P. Milbradt and T. Pick, Polytope finite elements, Int. J. Numer. Methods Eng., 73(12):1811- 1835, 2008

    work page 2008

  26. [26]

    Morley, The triangular equilibrium element in the solution of plate bending problems, Aero

    L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart., 19(2):149-169, 1968. 24 10-4 10-3 10-2 h* 102 104 107 = (2,1) Fig. 5.15 . Relation between maxx∈K |Dαϕi(x)| and h∗ for α = (2, 1) on K1. 10-3 10-2 h* 101 102 103 104 = (0,2),(0,3) 10-3 10-2 h* 101 102 = (0,2),(0,3) Fig. 5.16 . Relation between maxx∈K...

    work page 1968

  27. [27]

    A. Rand, A. Gillette and C. Bajaj, Interpolation error estimates for mean value coordinates over convex polygons, Adv. Comput. Math., 39:327-347, 2013

    work page 2013

  28. [28]

    A. Rand, A. Gillette and C. Bajaj, Quadratic serendipity finite elements on polygons using generalized barycentric coordinates, Math. Comp., 83:2691-2716, 2014

    work page 2014

  29. [29]

    Sieger, P

    D. Sieger, P. Alliez and M. Botsch, Optimizing Voronoi diagrams for polygonal finite element computations, Proc. 19th Int. Meshing Roundtable, 335-350, 2010

    work page 2010

  30. [30]

    Sukumar and E.A

    N. Sukumar and E.A. Malsch, Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Eng., 13(1):129-163, 2006

    work page 2006

  31. [31]

    Sukumar and A

    N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Int. J. Numer. Methods Eng., 61(12):2045-2066, 2004

    work page 2045

  32. [32]

    Tabarraei and N

    A. Tabarraei and N. Sukumar, Application of polygonal finite elements in linear elasticity, Int. J. Comput. Methods, 3(4):503-520, 2006

    work page 2006

  33. [33]

    Talischi, G.H

    C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struc. Multidisc. Optim., 45:309-328, 2012

    work page 2012

  34. [34]

    Tian, A quadratic Morley-type polygonal element and its application to plate bending prob- lem, Master’s Thesis (in Chinese), Nanjing Normal University, 2022

    P. Tian, A quadratic Morley-type polygonal element and its application to plate bending prob- lem, Master’s Thesis (in Chinese), Nanjing Normal University, 2022

    work page 2022

  35. [35]

    Wachspress, A Rational Finite Element Basis, Mathematics in Science and Engineering, vol

    E. Wachspress, A Rational Finite Element Basis, Mathematics in Science and Engineering, vol. 114, Academic Press, Inc. [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975

    work page 1975

  36. [36]

    Wachspress, Rational Bases and Generalized Barycentrics, Springer, New York, 2016

    E. Wachspress, Rational Bases and Generalized Barycentrics, Springer, New York, 2016

    work page 2016

  37. [37]

    Warren, Barycentric coordinates for convex polytopes, Adv

    J. Warren, Barycentric coordinates for convex polytopes, Adv. Comput. Math., 6:97-108, 1996

    work page 1996

  38. [38]

    Warren, S

    J. Warren, S. Schaefer, A. N. Hirani and M. Desbrun, Barycentric coordinates for convex sets, Adv. Comput. Math., 27:319-338, 2007

    work page 2007

  39. [39]

    Wicke, M

    M. Wicke, M. Botsch and M. Gross, A finite element method on convex polyhedra, Comput. Graph. Forum., 26(3):355-364, 2007. 25 10-3 10-2 h* 102 105 108 = (1,2) Fig. 5.17 . Relation between maxx∈K |Dαϕi(x)| and h∗ for α = (1, 2) on K2. 10-3 10-2 h* 101 102 103 Tangential derivatives 10-4 10-2 h* 100 101 102 Tangential derivatives Fig. 5.18 . Relation betwee...

    work page 2007

  40. [40]

    Yaglom and V.G

    I.M. Yaglom and V.G. Boltyanski, Convex figures, (Translated by P.J. Kelly and L.F. Walton), Holt, Rinehart and Winston, New York, 1961

    work page 1961

  41. [41]

    Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995

    G.M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995. 26 Fig. 5.19. Meshes of size 8 ×8. (1) A triangle mesh. (2) A rectangular mesh. (3) A pentagons mesh, with mostly pentagons and a few triangles. (4) A hexagonal mesh, with mostly hexagons and a few pentagons and quadrilaterals. (5) Centroidal Voronoi tessellation. 0.02 0.04 0.06 0.08 h ...

    work page 1995