An \(hp\) Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations.
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| Title: | An \(hp\) Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations. |
|---|---|
| Authors: | Margenberg, Nils1 (AUTHOR) nils.margenberg@ovgu.de, Bause, Markus2 (AUTHOR) bause@hsu-hh.de, Munch, Peter3 (AUTHOR) p.muench@tu-berlin.de |
| Source: | SIAM Journal on Scientific Computing. 2025, Vol. 47 Issue 6, pB1503-B1529. 27p. |
| Subjects: | Multigrid methods (Numerical analysis), Stokes equations, Galerkin methods, Finite element method, High performance computing |
| Abstract: | We present a monolithic \(hp\) space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs \(\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}\) of elements in space and a discontinuous Galerkin (DG \((k)\)) discretization in time with piecewise polynomials of order \(k\). The key novelty of this work is the application of \(hp\) multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal.II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \(hp\) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second. Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [ 126KB]). [ABSTRACT FROM AUTHOR] |
| Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics 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 |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 191459279 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: An \(hp\) Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Margenberg%2C+Nils%22">Margenberg, Nils</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> nils.margenberg@ovgu.de</i><br /><searchLink fieldCode="AR" term="%22Bause%2C+Markus%22">Bause, Markus</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> bause@hsu-hh.de</i><br /><searchLink fieldCode="AR" term="%22Munch%2C+Peter%22">Munch, Peter</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> p.muench@tu-berlin.de</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Scientific+Computing%22">SIAM Journal on Scientific Computing</searchLink>. 2025, Vol. 47 Issue 6, pB1503-B1529. 27p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Multigrid+methods+%28Numerical+analysis%29%22">Multigrid methods (Numerical analysis)</searchLink><br /><searchLink fieldCode="DE" term="%22Stokes+equations%22">Stokes equations</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22High+performance+computing%22">High performance computing</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present a monolithic \(hp\) space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs \(\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}\) of elements in space and a discontinuous Galerkin (DG \((k)\)) discretization in time with piecewise polynomials of order \(k\). The key novelty of this work is the application of \(hp\) multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal.II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \(hp\) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second. Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [ 126KB]). [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics 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: BibEntity: Identifiers: – Type: doi Value: 10.1137/25M1734142 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 27 StartPage: B1503 Subjects: – SubjectFull: Multigrid methods (Numerical analysis) Type: general – SubjectFull: Stokes equations Type: general – SubjectFull: Galerkin methods Type: general – SubjectFull: Finite element method Type: general – SubjectFull: High performance computing Type: general Titles: – TitleFull: An \(hp\) Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Margenberg, Nils – PersonEntity: Name: NameFull: Bause, Markus – PersonEntity: Name: NameFull: Munch, Peter IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: 2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 10648275 Numbering: – Type: volume Value: 47 – Type: issue Value: 6 Titles: – TitleFull: SIAM Journal on Scientific Computing Type: main |
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