Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs.
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| Title: | Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs. |
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| Authors: | Seyednazari, Salar1 (AUTHOR) salarseyednazari@gmail.com, Tatari, Mehdi1,2 (AUTHOR) mtatari@cc.iut.ac.ir, Mirzaei, Davoud3 (AUTHOR) d.mirzaei@sci.ui.ac.ir |
| Source: | Computers & Mathematics with Applications. Dec2021, Vol. 103, p1-11. 11p. |
| Subjects: | Elliptic differential equations, Differential operators, Sobolev spaces, Algebraic equations, Selfadjoint operators, Rayleigh model |
| Abstract: | We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds. [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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| Items | – Name: Title Label: Title Group: Ti Data: Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Seyednazari%2C+Salar%22">Seyednazari, Salar</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> salarseyednazari@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Tatari%2C+Mehdi%22">Tatari, Mehdi</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> mtatari@cc.iut.ac.ir</i><br /><searchLink fieldCode="AR" term="%22Mirzaei%2C+Davoud%22">Mirzaei, Davoud</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> d.mirzaei@sci.ui.ac.ir</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Dec2021, Vol. 103, p1-11. 11p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Elliptic+differential+equations%22">Elliptic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+operators%22">Differential operators</searchLink><br /><searchLink fieldCode="DE" term="%22Sobolev+spaces%22">Sobolev spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+equations%22">Algebraic equations</searchLink><br /><searchLink fieldCode="DE" term="%22Selfadjoint+operators%22">Selfadjoint operators</searchLink><br /><searchLink fieldCode="DE" term="%22Rayleigh+model%22">Rayleigh model</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds. [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.2021.10.019 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 11 StartPage: 1 Subjects: – SubjectFull: Elliptic differential equations Type: general – SubjectFull: Differential operators Type: general – SubjectFull: Sobolev spaces Type: general – SubjectFull: Algebraic equations Type: general – SubjectFull: Selfadjoint operators Type: general – SubjectFull: Rayleigh model Type: general Titles: – TitleFull: Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Seyednazari, Salar – PersonEntity: Name: NameFull: Tatari, Mehdi – PersonEntity: Name: NameFull: Mirzaei, Davoud IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2021 Type: published Y: 2021 Identifiers: – Type: issn-print Value: 08981221 Numbering: – Type: volume Value: 103 Titles: – TitleFull: Computers & Mathematics with Applications Type: main |
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