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An Algebraic Preconditioner for the Exactly Divergence‐Free Discontinuous Galerkin Method for Stokes.

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Title: An Algebraic Preconditioner for the Exactly Divergence‐Free Discontinuous Galerkin Method for Stokes.
Authors: Rhebergen, Sander1 (AUTHOR) srheberg@uwaterloo.ca, Southworth, Ben S.2 (AUTHOR)
Source: Numerical Methods for Partial Differential Equations. Mar2025, Vol. 41 Issue 2, p1-10. 10p.
Subjects: Algebraic multigrid methods, Stokes equations, Schur complement, Finite element method, Galerkin methods
Abstract: We present an algebraic preconditioner for the exactly divergence‐free discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and Schötzau [J. Sci. Comput., 31 (2007), pp. 61–73] and Wang and Ye [SIAM J. Numer. Anal., 45 (2007), pp. 1269–1286] for the Stokes problem. The exactly divergence‐free DG method uses finite elements that use an H(div)$$ H\left(\operatorname{div}\right) $$‐conforming basis, thereby significantly complicating its solution by iterative methods. Several preconditioners for this Stokes discretization has been developed, but each is based on specialized solvers or decompositions. To avoid requiring custom solvers, we hybridize the H(div)$$ H\left(\operatorname{div}\right) $$‐conforming finite element so that the velocity lives in a standard L2$$ {L}^2 $$‐DG space, and present a simple algebraic preconditioner for the extended hybridized system. The proposed preconditioner is optimal in mesh size h$$ h $$, effective in 2d and 3d, and only relies on standard relaxation and algebraic multigrid methods available in many packages. Furthermore, the Schur complement approximation is robust in element order k$$ k $$, although more AMG cycles are needed on the velocity block when increasing k$$ k $$. [ABSTRACT FROM AUTHOR]
Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell 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: An Algebraic Preconditioner for the Exactly Divergence‐Free Discontinuous Galerkin Method for Stokes.
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  Data: <searchLink fieldCode="AR" term="%22Rhebergen%2C+Sander%22">Rhebergen, Sander</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> srheberg@uwaterloo.ca</i><br /><searchLink fieldCode="AR" term="%22Southworth%2C+Ben+S%2E%22">Southworth, Ben S.</searchLink><relatesTo>2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Numerical+Methods+for+Partial+Differential+Equations%22">Numerical Methods for Partial Differential Equations</searchLink>. Mar2025, Vol. 41 Issue 2, p1-10. 10p.
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  Data: <searchLink fieldCode="DE" term="%22Algebraic+multigrid+methods%22">Algebraic multigrid methods</searchLink><br /><searchLink fieldCode="DE" term="%22Stokes+equations%22">Stokes equations</searchLink><br /><searchLink fieldCode="DE" term="%22Schur+complement%22">Schur complement</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink>
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  Data: We present an algebraic preconditioner for the exactly divergence‐free discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and Schötzau [J. Sci. Comput., 31 (2007), pp. 61–73] and Wang and Ye [SIAM J. Numer. Anal., 45 (2007), pp. 1269–1286] for the Stokes problem. The exactly divergence‐free DG method uses finite elements that use an H(div)$$ H\left(\operatorname{div}\right) $$‐conforming basis, thereby significantly complicating its solution by iterative methods. Several preconditioners for this Stokes discretization has been developed, but each is based on specialized solvers or decompositions. To avoid requiring custom solvers, we hybridize the H(div)$$ H\left(\operatorname{div}\right) $$‐conforming finite element so that the velocity lives in a standard L2$$ {L}^2 $$‐DG space, and present a simple algebraic preconditioner for the extended hybridized system. The proposed preconditioner is optimal in mesh size h$$ h $$, effective in 2d and 3d, and only relies on standard relaxation and algebraic multigrid methods available in many packages. Furthermore, the Schur complement approximation is robust in element order k$$ k $$, although more AMG cycles are needed on the velocity block when increasing k$$ k $$. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell 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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        Value: 10.1002/num.70001
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      – Code: eng
        Text: English
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        PageCount: 10
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    Subjects:
      – SubjectFull: Algebraic multigrid methods
        Type: general
      – SubjectFull: Stokes equations
        Type: general
      – SubjectFull: Schur complement
        Type: general
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Galerkin methods
        Type: general
    Titles:
      – TitleFull: An Algebraic Preconditioner for the Exactly Divergence‐Free Discontinuous Galerkin Method for Stokes.
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            NameFull: Rhebergen, Sander
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              M: 03
              Text: Mar2025
              Type: published
              Y: 2025
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              Value: 41
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            – TitleFull: Numerical Methods for Partial Differential Equations
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