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arxiv: 2606.24423 · v1 · pith:F3NUXP7Rnew · submitted 2026-06-23 · 🧮 math.NA · cs.NA

A sparse spectral method on a class of domains bounded by planar algebraic curves

Jiajie Yao , Marco Fasondini , Sheehan Olver This is my paper

Pith reviewed 2026-06-25 23:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords spectral methodsparse discretizationKoornwinder polynomialsalgebraic curvesorthogonal polynomialsnumerical PDEdomain discretizationsemiclassical polynomials
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The pith

Generalised bivariate Koornwinder polynomials built from semiclassical orthogonal polynomials yield sparse spectral discretizations for PDEs on domains bounded by planar algebraic curves.

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

This paper develops a spectral method for solving partial differential equations on two-dimensional domains whose boundaries consist of planar algebraic curves. The approach relies on generalised bivariate Koornwinder polynomials that form a complete orthogonal basis on these domains, even though the basis is not graded by polynomial degree. These polynomials are assembled from new families of univariate semiclassical orthogonal polynomials, for which the Jacobi, raising and differentiation matrices are constructed with linear complexity in the number of functions. The resulting discretizations produce sparse matrices, and fast transforms convert between grid values and expansion coefficients. Experiments on smooth, piecewise-smooth and non-convex geometries show algebraic convergence rates that become exponential once the boundary is smooth.

Core claim

The central claim is that generalised bivariate Koornwinder polynomials, constructed from univariate semiclassical orthogonal polynomials whose associated operator matrices are computed with optimal linear complexity, furnish a complete orthogonal basis on the target domains; when these polynomials are used to discretise PDEs the resulting matrices are sparse and admit fast transforms from grid values to coefficients.

What carries the argument

generalised bivariate Koornwinder polynomials formed from univariate semiclassical orthogonal polynomials; these supply the complete orthogonal basis whose operator matrices remain sparse after discretization.

If this is right

  • PDE discretizations on the described domains produce sparse matrices that support efficient direct or iterative solvers.
  • Fast, stable transforms exist between values on a grid and the coefficients in the Koornwinder expansion.
  • Convergence is algebraic when the boundary contains corners and becomes spectral once the boundary is smooth.
  • The same construction applies without modification to non-convex geometries whose boundaries remain algebraic curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The linear-complexity matrix construction may extend the method to problems with thousands of degrees of freedom without requiring mesh generation.
  • Because the basis is orthogonal but not degree-graded, similar constructions could be explored for other families of orthogonal polynomials on non-standard domains.
  • The observed transition from algebraic to spectral convergence with boundary smoothness suggests the method could serve as a diagnostic tool for detecting geometric regularity in inverse problems.

Load-bearing premise

The generalised bivariate Koornwinder polynomials form a complete orthogonal basis on the target domains and the univariate semiclassical operator matrices can be computed with optimal linear complexity.

What would settle it

Demonstrating that the bivariate polynomials fail to span the function space on a given algebraic domain, or that the discretisation matrices lose sparsity for any smooth test problem, would falsify the central claim.

read the original abstract

We develop a sparse spectral method for solving partial differential equations on a class of two-dimensional geometries bounded by algebraic curves. The numerical method uses generalised bivariate Koornwinder polynomials which form a complete orthogonal basis, but one which is not graded in terms of polynomial degree. The polynomials are built from new families of univariate semiclassical orthogonal polynomials whose associated operator matrices (Jacobi matrices, raising matrices and differentiation matrices) are computed with optimal linear complexity in the number of basis functions. When used to discretise partial differential equations the resulting matrices are sparse enabling efficient numerical solution. Moreover, we develop fast transforms from values on a grid to expansion coefficients. The efficiency and accuracy of the resulting spectral method are illustrated through a series of numerical experiments on geometries whose boundaries are smooth and piecewise smooth including non-convex geometries. We observe algebraic convergence for geometries with corners, which accelerates to exponentially fast (spectral) convergence when the boundary is smooth.

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

2 major / 2 minor

Summary. The manuscript develops a sparse spectral method for PDEs on 2D domains bounded by planar algebraic curves. It constructs generalised bivariate Koornwinder polynomials from new families of univariate semiclassical orthogonal polynomials, for which Jacobi, raising, and differentiation matrices are computed in linear complexity. The resulting discretizations are claimed to be sparse, with accompanying fast transforms between grid values and coefficients. Numerical experiments on smooth, piecewise-smooth, and non-convex geometries illustrate algebraic convergence at corners accelerating to spectral convergence on smooth boundaries.

Significance. If the orthogonality and completeness claims hold, the method supplies an efficient, sparsity-exploiting spectral discretization for a useful class of non-standard domains, with the linear-complexity matrix construction and fast transforms as concrete strengths. The observed convergence rates on non-convex and piecewise-smooth boundaries would be a useful addition to the literature on spectral methods for irregular geometries.

major comments (2)
  1. [Abstract and §2] Abstract and §2 (construction of the bivariate basis): the claim that the generalised bivariate Koornwinder polynomials form a complete orthogonal basis (hence identity mass matrix) on domains bounded by arbitrary planar algebraic curves rests on an extension of the univariate semiclassical construction. No explicit verification is given that the induced inner product on a general (possibly non-convex) algebraic curve remains diagonal after the tensoring or combination step; if the measure does not separate or satisfy the required semiclassical ODE, the sparsity of the discretised operators collapses.
  2. [§4] §4 (discretisation of PDEs): the sparsity of the resulting matrices is asserted to follow directly from the univariate operator matrices being sparse. This is load-bearing for the central efficiency claim, yet the argument does not address how the bivariate inner-product measure induced by a non-rectangular algebraic domain preserves the sparsity pattern when the boundary is only piecewise algebraic.
minor comments (2)
  1. [Abstract] The abstract states 'we observe algebraic convergence for geometries with corners'; a brief remark on the expected rate (e.g., O(N^{-r}) for corner angle r) would clarify the experiments.
  2. [§2] Notation for the univariate semiclassical families is introduced without a compact summary table relating the parameters of the algebraic curve to the semiclassical weight; adding such a table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for stronger justification of the orthogonality and sparsity claims. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (construction of the bivariate basis): the claim that the generalised bivariate Koornwinder polynomials form a complete orthogonal basis (hence identity mass matrix) on domains bounded by arbitrary planar algebraic curves rests on an extension of the univariate semiclassical construction. No explicit verification is given that the induced inner product on a general (possibly non-convex) algebraic curve remains diagonal after the tensoring or combination step; if the measure does not separate or satisfy the required semiclassical ODE, the sparsity of the discretised operators collapses.

    Authors: The univariate semiclassical families are constructed to satisfy the ODE induced by the algebraic curve equation, so that the resulting bivariate combination yields an orthogonal basis (identity mass matrix) with respect to the standard area measure on the domain. Completeness holds because the polynomials are dense in L2(D) for any bounded D whose boundary is an algebraic curve. The construction extends directly to non-convex domains because the inner product is the restriction of Lebesgue measure and does not require convexity. We agree that an explicit verification paragraph is missing and will add it to §2 (and update the abstract) showing how the semiclassical property is preserved under the bivariate step. revision: yes

  2. Referee: [§4] §4 (discretisation of PDEs): the sparsity of the resulting matrices is asserted to follow directly from the univariate operator matrices being sparse. This is load-bearing for the central efficiency claim, yet the argument does not address how the bivariate inner-product measure induced by a non-rectangular algebraic domain preserves the sparsity pattern when the boundary is only piecewise algebraic.

    Authors: Sparsity is preserved because the bivariate differentiation and multiplication operators are assembled via Kronecker-type combinations of the univariate sparse matrices; the identity mass matrix introduces no additional fill-in. For piecewise-algebraic boundaries the same global basis is used, and the operator matrices remain sparse because the univariate sparsity pattern is inherited regardless of how the domain is partitioned. We will revise §4 to include an explicit statement and a short schematic of the sparsity pattern for the piecewise case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper premises that generalised bivariate Koornwinder polynomials form a complete orthogonal basis on the target domains and builds sparse discretisations from univariate semiclassical operator matrices whose linear-complexity construction is independent of the target PDE solution. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation chain by construction. The abstract and stated assumptions treat orthogonality and completeness as given properties of the chosen basis rather than outputs derived from the numerical method itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

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Works this paper leans on

25 extracted references · 19 canonical work pages

  1. [1]

    C. W. Clenshaw. A note on the summation of Chebyshev series.Math. Comp., 9(51):118– 120, 1955.doi:10.1090/s0025-5718-1955-0071856-0. 32 Decay of expansion coefficients of solutions to the biharmonic equation FIGURE 7Left: The approximate solution to ∆ 2u=f, subject to zero Dirichlet and Neumann conditions, withf(x, y) = 10 4 sin(30πx) cos(30πy). Right: th...

    work page doi:10.1090/s0025-5718-1955-0071856-0 1955

  2. [2]

    C. F. Dunkl and Y. Xu.Orthogonal Polynomials of Several Variables. Number 155 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2014.doi: 10.1017/cbo9781107786134

    work page doi:10.1017/cbo9781107786134 2014

  3. [3]

    A. C. Ellison and K. Julien. Gyroscopic polynomials.J. Comput. Phys., 489:112268, 2023. doi:10.1016/j.jcp.2023.112268

    work page doi:10.1016/j.jcp.2023.112268 2023

  4. [4]

    Grisvard.Elliptic Problems in Nonsmooth Domains

    P. Grisvard.Elliptic Problems in Nonsmooth Domains. SIAM, 2011

    2011

  5. [5]

    Gumerov, S

    K. Gumerov, S. Rigg, and R. M. Slevinsky. Fast measure modification of orthogonal poly- nomials via matrices with displacement structure, 2024. URL:https://arxiv.org/abs/ 2412.17663,arXiv:2412.17663

    work page arXiv 2024

  6. [6]

    T. S. Gutleb, S. Olver, and R. M. Slevinsky. Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations.Found. Comput. Math., pages 1–43, 2024.doi:10.1007/s10208-024-09671-w

    work page doi:10.1007/s10208-024-09671-w 2024

  7. [7]

    Hendriksen and H

    E. Hendriksen and H. van Rossum. Semi-classical orthogonal polynomials. InPolynˆ omes Orthogonaux et Applications: Proceedings of the Laguerre Symposium Held at Bar-le-Duc, October 15–18, 1984, pages 354–361. Springer, 2006.doi:10.1007/bfb0076564

    work page doi:10.1007/bfb0076564 1984

  8. [8]

    L. Jia, H. Li, and Z. Zhang. Sparse spectral-Galerkin method on an arbitrary tetrahedron using generalized Koornwinder polynomials.J. Sci. Comput., 91(1):22, 2022.doi:10. 1007/s10915-022-01778-y. 33

    2022

  9. [9]

    Komatitsch, C

    D. Komatitsch, C. Barnes, and J. Tromp. Simulation of anisotropic wave propagation based upon a spectral element method.Geophysics, 65(4):1251–1260, 2000.doi:10.1190/ 1.1444816

    2000

  10. [10]

    Koornwinder

    T. Koornwinder. Two-variable analogues of the classical orthogonal polynomials. InThe- ory and Application of Special Functions, pages 435–495. Elsevier, 1975.doi:10.1016/ b978-0-12-064850-4.50015-x

    1975

  11. [11]

    A. P. Magnus. Painlev´ e-type differential equations for the recurrence coefficients of semi- classical orthogonal polynomials.J. Comput. Appl. Math., 57(1-2):215–237, 1995.doi: 10.1016/0377-0427(93)E0247-J

    work page doi:10.1016/0377-0427(93)e0247-j 1995

  12. [12]

    Marty, C

    P. Marty, C. Boehm, M. van Driel, and A. Fichtner. Transcranial ultrasound modeling using the spectral-element method.J. Acoust. Soc. Am., 156(6):3674–3693, 2024.doi: 10.1121/10.0034474

    work page doi:10.1121/10.0034474 2024

  13. [13]

    F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15, 2025. URL: http://dlmf.nist.gov/

    2025

  14. [14]

    Olver, R

    S. Olver, R. M. Slevinsky, and A. Townsend. Fast algorithms using orthogonal polynomials. Acta Numer., 29:573–699, 2020.doi:10.1017/s0962492920000045

    work page doi:10.1017/s0962492920000045 2020

  15. [15]

    Olver, A

    S. Olver, A. Townsend, and G. Vasil. A sparse spectral method on triangles.SIAM J. Sci. Comput., 41(6):A3728–A3756, 2019.doi:10.1137/19m1245888

    work page doi:10.1137/19m1245888 2019

  16. [16]

    I. P. Papadopoulos, T. S. Gutleb, R. M. Slevinsky, and S. Olver. Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli.SIAM J. Sci. Comput., 46(6):A3448–A3476, 2024.doi:10.1137/23m160846x

    work page doi:10.1137/23m160846x 2024

  17. [17]

    R. M. Slevinsky. On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev–Jacobi transform.IMA J. Numer. Anal., 38(1):102–124, 2018.doi:10.1093/imanum/drw070

    work page doi:10.1093/imanum/drw070 2018

  18. [18]

    R. M. Slevinsky. Fast and backward stable transforms between spherical harmonic expan- sions and bivariate Fourier series.Appl. Comput. Harmon. Anal., 47(3):585–606, 2019. doi:10.1016/j.acha.2017.11.001

    work page doi:10.1016/j.acha.2017.11.001 2019

  19. [19]

    F. J. Smith. An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation.Math. Comp., 19(89):33–36, 1965. doi:10.2307/2004093

    work page doi:10.2307/2004093 1965

  20. [20]

    Snowball and S

    B. Snowball and S. Olver. Sparse spectral andp-finite element methods for partial dif- ferential equations on disk slices and trapeziums.Stud. Appl. Math., 145(1):3–35, 2020. doi:10.1111/sapm.12303

    work page doi:10.1111/sapm.12303 2020

  21. [21]

    Snowball and S

    B. Snowball and S. Olver. Sparse spectral methods for partial differential equations on spherical caps.Trans. Math. Appl., 5(1):tnab001, 2021.doi:10.1093/imatrm/tnab001. 34

    work page doi:10.1093/imatrm/tnab001 2021

  22. [22]

    Staniforth and J

    A. Staniforth and J. Thuburn. Horizontal grids for global weather and climate prediction models: a review.Q. J. R. Meteorol. Soc., 138(662):1–26, 2012.doi:10.1002/qj.958

    work page doi:10.1002/qj.958 2012

  23. [23]

    VandenHeuvel and S

    D. VandenHeuvel and S. Olver. A sparsehp-finite element method for piecewise-smooth differential equations with periodic boundary conditions.Numer. Algorithms, pages 1–61, 2026.doi:10.1007/s11075-026-02313-y

    work page doi:10.1007/s11075-026-02313-y 2026

  24. [24]

    G. M. Vasil, K. J. Burns, D. Lecoanet, S. Olver, B. P. Brown, and J. S. Oishi. Tensor calculus in polar coordinates using Jacobi polynomials.J. Comput. Phys., 325:53–73, 2016. doi:10.1016/j.jcp.2016.08.013

    work page doi:10.1016/j.jcp.2016.08.013 2016

  25. [25]

    J. Yao. Generalisedkoornwinderpolynomials.jl, 2025. URL:https://github.com/ JiajieYao1106/GeneralisedKoornwinderPolynomials.jl. A. Generalised Koornwinder polynomials on smooth do- mains It can be shown that for a generalised Koornwinder domain (1.3) to be smooth,ρmust take the form ρ= p (x−α)(β−x)ϕ(x), x∈[α, β], whereϕis a polynomial satisfyingϕ(x)>0 for...

    2025