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On the Unisolvence for the Quasi-Polynomial Spaces of Differential Forms.

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Title: On the Unisolvence for the Quasi-Polynomial Spaces of Differential Forms.
Authors: Wu, Shuonan1 (AUTHOR) snwu@math.pku.edu.cn, Zikatanov, Ludmil T.2 (AUTHOR) ludmil@psu.edu
Source: Computational Methods in Applied Mathematics. Apr2026, Vol. 26 Issue 2, p285-307. 23p.
Subjects: Differential forms, Polynomials, Sobolev spaces, Equations, Advection-diffusion equations, Tetrahedra, Uniqueness (Mathematics), Function spaces
Abstract: We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes. Numerical experiments for scalar convection-diffusion problems in 2D demonstrate the superior performance of the exponentially-fitted method based on 풫 2 -Lagrange elements. [ABSTRACT FROM AUTHOR]
Copyright of Computational Methods in Applied Mathematics is the property of De Gruyter and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: On the Unisolvence for the Quasi-Polynomial Spaces of Differential Forms.
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  Data: <searchLink fieldCode="AR" term="%22Wu%2C+Shuonan%22">Wu, Shuonan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> snwu@math.pku.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Zikatanov%2C+Ludmil+T%2E%22">Zikatanov, Ludmil T.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> ludmil@psu.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Computational+Methods+in+Applied+Mathematics%22">Computational Methods in Applied Mathematics</searchLink>. Apr2026, Vol. 26 Issue 2, p285-307. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Differential+forms%22">Differential forms</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Sobolev+spaces%22">Sobolev spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Equations%22">Equations</searchLink><br /><searchLink fieldCode="DE" term="%22Advection-diffusion+equations%22">Advection-diffusion equations</searchLink><br /><searchLink fieldCode="DE" term="%22Tetrahedra%22">Tetrahedra</searchLink><br /><searchLink fieldCode="DE" term="%22Uniqueness+%28Mathematics%29%22">Uniqueness (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Function+spaces%22">Function spaces</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes. Numerical experiments for scalar convection-diffusion problems in 2D demonstrate the superior performance of the exponentially-fitted method based on 풫 2 -Lagrange elements. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Computational Methods in Applied Mathematics is the property of De Gruyter and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1515/cmam-2025-0135
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 23
        StartPage: 285
    Subjects:
      – SubjectFull: Differential forms
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Sobolev spaces
        Type: general
      – SubjectFull: Equations
        Type: general
      – SubjectFull: Advection-diffusion equations
        Type: general
      – SubjectFull: Tetrahedra
        Type: general
      – SubjectFull: Uniqueness (Mathematics)
        Type: general
      – SubjectFull: Function spaces
        Type: general
    Titles:
      – TitleFull: On the Unisolvence for the Quasi-Polynomial Spaces of Differential Forms.
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            NameFull: Wu, Shuonan
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            – D: 01
              M: 04
              Text: Apr2026
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              Y: 2026
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            – TitleFull: Computational Methods in Applied Mathematics
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