A posteriori error estimates for the finite element approximation of the convection–diffusion–reaction equation based on the variational multiscale concept.
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| Title: | A posteriori error estimates for the finite element approximation of the convection–diffusion–reaction equation based on the variational multiscale concept. |
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| Authors: | Codina, Ramon1,2 (AUTHOR) ramon.codina@upc.edu, Gravenkamp, Hauke3 (AUTHOR) hauke.gravenkamp@ovgu.de, Khan, Sheraz Ahmed1,2 (AUTHOR) sheraz.ahmed@upc.edu |
| Source: | Applied Numerical Mathematics. Dec2025, Vol. 218, p238-260. 23p. |
| Subjects: | A posteriori error analysis, Transport equation, A priori, Equations |
| Abstract: | In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family. [ABSTRACT FROM AUTHOR] |
| Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. 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: 188153521 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A posteriori error estimates for the finite element approximation of the convection–diffusion–reaction equation based on the variational multiscale concept. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Codina%2C+Ramon%22">Codina, Ramon</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> ramon.codina@upc.edu</i><br /><searchLink fieldCode="AR" term="%22Gravenkamp%2C+Hauke%22">Gravenkamp, Hauke</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> hauke.gravenkamp@ovgu.de</i><br /><searchLink fieldCode="AR" term="%22Khan%2C+Sheraz+Ahmed%22">Khan, Sheraz Ahmed</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> sheraz.ahmed@upc.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Applied+Numerical+Mathematics%22">Applied Numerical Mathematics</searchLink>. Dec2025, Vol. 218, p238-260. 23p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22A+posteriori+error+analysis%22">A posteriori error analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Transport+equation%22">Transport equation</searchLink><br /><searchLink fieldCode="DE" term="%22A+priori%22">A priori</searchLink><br /><searchLink fieldCode="DE" term="%22Equations%22">Equations</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. 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.apnum.2025.08.003 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 23 StartPage: 238 Subjects: – SubjectFull: A posteriori error analysis Type: general – SubjectFull: Transport equation Type: general – SubjectFull: A priori Type: general – SubjectFull: Equations Type: general Titles: – TitleFull: A posteriori error estimates for the finite element approximation of the convection–diffusion–reaction equation based on the variational multiscale concept. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Codina, Ramon – PersonEntity: Name: NameFull: Gravenkamp, Hauke – PersonEntity: Name: NameFull: Khan, Sheraz Ahmed IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01689274 Numbering: – Type: volume Value: 218 Titles: – TitleFull: Applied Numerical Mathematics Type: main |
| ResultId | 1 |
