Analysis of the weighted shifted boundary method for the Poisson and Stokes problems.
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| Title: | Analysis of the weighted shifted boundary method for the Poisson and Stokes problems. |
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| Authors: | Atallah, Nabil M.1 (AUTHOR) atallah1@llnl.gov, Canuto, Claudio1,2 (AUTHOR) claudio.canuto@polito.it, Scovazzi, Guglielmo3 (AUTHOR) guglielmo.scovazzi@duke.edu |
| Source: | Computers & Mathematics with Applications. Mar2026, Vol. 205, p63-85. 23p. |
| Subjects: | Poisson's equation, Stokes equations, Stability theory, Numerical analysis, Finite element method, Stability (Mechanics) |
| Abstract: | The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute "weighted" in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L 2-error estimates can also be derived. [ABSTRACT FROM AUTHOR] |
| Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science 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: 191449131 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Analysis of the weighted shifted boundary method for the Poisson and Stokes problems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Atallah%2C+Nabil+M%2E%22">Atallah, Nabil M.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> atallah1@llnl.gov</i><br /><searchLink fieldCode="AR" term="%22Canuto%2C+Claudio%22">Canuto, Claudio</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> claudio.canuto@polito.it</i><br /><searchLink fieldCode="AR" term="%22Scovazzi%2C+Guglielmo%22">Scovazzi, Guglielmo</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> guglielmo.scovazzi@duke.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Mar2026, Vol. 205, p63-85. 23p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Poisson's+equation%22">Poisson's equation</searchLink><br /><searchLink fieldCode="DE" term="%22Stokes+equations%22">Stokes equations</searchLink><br /><searchLink fieldCode="DE" term="%22Stability+theory%22">Stability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Stability+%28Mechanics%29%22">Stability (Mechanics)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute "weighted" in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L 2-error estimates can also be derived. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science 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: 10.1016/j.camwa.2025.12.011 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 23 StartPage: 63 Subjects: – SubjectFull: Poisson's equation Type: general – SubjectFull: Stokes equations Type: general – SubjectFull: Stability theory Type: general – SubjectFull: Numerical analysis Type: general – SubjectFull: Finite element method Type: general – SubjectFull: Stability (Mechanics) Type: general Titles: – TitleFull: Analysis of the weighted shifted boundary method for the Poisson and Stokes problems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Atallah, Nabil M. – PersonEntity: Name: NameFull: Canuto, Claudio – PersonEntity: Name: NameFull: Scovazzi, Guglielmo IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Text: Mar2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 08981221 Numbering: – Type: volume Value: 205 Titles: – TitleFull: Computers & Mathematics with Applications Type: main |
| ResultId | 1 |
