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Stability of conforming space--time isogeometric methods for the wave equation.

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Title: Stability of conforming space--time isogeometric methods for the wave equation.
Authors: Ferrari, Matteo1 (AUTHOR), Fraschini, Sara2 (AUTHOR)
Source: Mathematics of Computation. Mar2026, Vol. 95 Issue 358, p683-719. 37p.
Subjects: Stability criterion, Wave equation, Numerical analysis, Dynamic stability, Finite element method, Mathematical regularization, Splines, Isogeometric analysis
Abstract: We consider a family of conforming space–time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a Courant–Friedrichs–Lewy (CFL) condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2019), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2024), have introduced unconditionally stable schemes by adding nonconsistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Computation is the property of American Mathematical Society 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.)
Database: Engineering Source
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Header DbId: egs
DbLabel: Engineering Source
An: 189912924
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PubTypeId: academicJournal
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  Data: Stability of conforming space--time isogeometric methods for the wave equation.
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  Data: <searchLink fieldCode="AR" term="%22Ferrari%2C+Matteo%22">Ferrari, Matteo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Fraschini%2C+Sara%22">Fraschini, Sara</searchLink><relatesTo>2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Mar2026, Vol. 95 Issue 358, p683-719. 37p.
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  Data: <searchLink fieldCode="DE" term="%22Stability+criterion%22">Stability criterion</searchLink><br /><searchLink fieldCode="DE" term="%22Wave+equation%22">Wave equation</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamic+stability%22">Dynamic stability</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+regularization%22">Mathematical regularization</searchLink><br /><searchLink fieldCode="DE" term="%22Splines%22">Splines</searchLink><br /><searchLink fieldCode="DE" term="%22Isogeometric+analysis%22">Isogeometric analysis</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We consider a family of conforming space–time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a Courant–Friedrichs–Lewy (CFL) condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2019), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2024), have introduced unconditionally stable schemes by adding nonconsistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society 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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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1090/mcom/4062
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 37
        StartPage: 683
    Subjects:
      – SubjectFull: Stability criterion
        Type: general
      – SubjectFull: Wave equation
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Dynamic stability
        Type: general
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Mathematical regularization
        Type: general
      – SubjectFull: Splines
        Type: general
      – SubjectFull: Isogeometric analysis
        Type: general
    Titles:
      – TitleFull: Stability of conforming space--time isogeometric methods for the wave equation.
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            NameFull: Ferrari, Matteo
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            NameFull: Fraschini, Sara
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          Dates:
            – D: 01
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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              Value: 95
            – Type: issue
              Value: 358
          Titles:
            – TitleFull: Mathematics of Computation
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