Multigrid defect correction and fourth-order compact scheme for Poisson’s equation.
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| Title: | Multigrid defect correction and fourth-order compact scheme for Poisson’s equation. |
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| Authors: | Abide, Stéphane1 stephane.abide@univ-perp.fr, Zeghmati, Belkacem1 |
| Source: | Computers & Mathematics with Applications. Apr2017, Vol. 73 Issue 7, p1433-1444. 12p. |
| Subjects: | Defect correction methods (Numerical analysis), Poisson's equation, Multigrid methods (Numerical analysis), Computational fluid dynamics, Mathematical decoupling, Neumann boundary conditions, Eigenvalues |
| Abstract: | This paper presents an analysis of a multigrid defect correction to solve a fourth-order compact scheme discretization of the Poisson’s equation. We focus on the formulation, which arises in the velocity/pressure decoupling methods encountered in computational fluid dynamics. Especially, the Poisson’s equation results of the divergence/gradient formulation and Neumann boundary conditions are prescribed. The convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix. The stability and the mesh-independency are demonstrated. An improvement of the convergence rate is suggested by introducing the damped Jacobi and Incomplete Lower Upper smoothers. Based on an eigenvalues analysis, the optimal damping parameter is proposed for each smoother. Numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions. Further numerical investigations allow us to extend the results of the eigenvalues analysis to Neumann boundary conditions and non-uniform meshes. The Hodge–Helmholtz decomposition of a vector field is carried out to illustrate the computational efficiency, especially by making comparisons with a second-order discretization of the Poisson’s equation solved with a state of art of algebraic multigrid method. [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: 121785002 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Multigrid defect correction and fourth-order compact scheme for Poisson’s equation. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Abide%2C+Stéphane%22">Abide, Stéphane</searchLink><relatesTo>1</relatesTo><i> stephane.abide@univ-perp.fr</i><br /><searchLink fieldCode="AR" term="%22Zeghmati%2C+Belkacem%22">Zeghmati, Belkacem</searchLink><relatesTo>1</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Apr2017, Vol. 73 Issue 7, p1433-1444. 12p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Defect+correction+methods+%28Numerical+analysis%29%22">Defect correction methods (Numerical analysis)</searchLink><br /><searchLink fieldCode="DE" term="%22Poisson's+equation%22">Poisson's equation</searchLink><br /><searchLink fieldCode="DE" term="%22Multigrid+methods+%28Numerical+analysis%29%22">Multigrid methods (Numerical analysis)</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+fluid+dynamics%22">Computational fluid dynamics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+decoupling%22">Mathematical decoupling</searchLink><br /><searchLink fieldCode="DE" term="%22Neumann+boundary+conditions%22">Neumann boundary conditions</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper presents an analysis of a multigrid defect correction to solve a fourth-order compact scheme discretization of the Poisson’s equation. We focus on the formulation, which arises in the velocity/pressure decoupling methods encountered in computational fluid dynamics. Especially, the Poisson’s equation results of the divergence/gradient formulation and Neumann boundary conditions are prescribed. The convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix. The stability and the mesh-independency are demonstrated. An improvement of the convergence rate is suggested by introducing the damped Jacobi and Incomplete Lower Upper smoothers. Based on an eigenvalues analysis, the optimal damping parameter is proposed for each smoother. Numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions. Further numerical investigations allow us to extend the results of the eigenvalues analysis to Neumann boundary conditions and non-uniform meshes. The Hodge–Helmholtz decomposition of a vector field is carried out to illustrate the computational efficiency, especially by making comparisons with a second-order discretization of the Poisson’s equation solved with a state of art of algebraic multigrid method. [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.2017.01.016 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 12 StartPage: 1433 Subjects: – SubjectFull: Defect correction methods (Numerical analysis) Type: general – SubjectFull: Poisson's equation Type: general – SubjectFull: Multigrid methods (Numerical analysis) Type: general – SubjectFull: Computational fluid dynamics Type: general – SubjectFull: Mathematical decoupling Type: general – SubjectFull: Neumann boundary conditions Type: general – SubjectFull: Eigenvalues Type: general Titles: – TitleFull: Multigrid defect correction and fourth-order compact scheme for Poisson’s equation. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Abide, Stéphane – PersonEntity: Name: NameFull: Zeghmati, Belkacem IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2017 Type: published Y: 2017 Identifiers: – Type: issn-print Value: 08981221 Numbering: – Type: volume Value: 73 – Type: issue Value: 7 Titles: – TitleFull: Computers & Mathematics with Applications Type: main |
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