Numerical solution of the boundary value problem of elliptic equation by Levi function scheme.
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| Title: | Numerical solution of the boundary value problem of elliptic equation by Levi function scheme. |
|---|---|
| Authors: | Pan, Jinchao1,2 (AUTHOR), Liu, Jijun1,2 (AUTHOR) jjliu@seu.edu.cn |
| Source: | Numerical Methods for Partial Differential Equations. Nov2024, Vol. 40 Issue 6, p1-26. 26p. |
| Subjects: | Numerical solutions to boundary value problems, Boundary value problems, Radial basis functions, Elliptic equations, Inhomogeneous materials |
| Abstract: | For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes. [ABSTRACT FROM AUTHOR] |
| Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell 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.) | |
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| Header | DbId: egs DbLabel: Engineering Source An: 179790068 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Numerical solution of the boundary value problem of elliptic equation by Levi function scheme. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Pan%2C+Jinchao%22">Pan, Jinchao</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Liu%2C+Jijun%22">Liu, Jijun</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> jjliu@seu.edu.cn</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Numerical+Methods+for+Partial+Differential+Equations%22">Numerical Methods for Partial Differential Equations</searchLink>. Nov2024, Vol. 40 Issue 6, p1-26. 26p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Numerical+solutions+to+boundary+value+problems%22">Numerical solutions to boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Radial+basis+functions%22">Radial basis functions</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+equations%22">Elliptic equations</searchLink><br /><searchLink fieldCode="DE" term="%22Inhomogeneous+materials%22">Inhomogeneous materials</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell 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: 10.1002/num.23142 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 26 StartPage: 1 Subjects: – SubjectFull: Numerical solutions to boundary value problems Type: general – SubjectFull: Boundary value problems Type: general – SubjectFull: Radial basis functions Type: general – SubjectFull: Elliptic equations Type: general – SubjectFull: Inhomogeneous materials Type: general Titles: – TitleFull: Numerical solution of the boundary value problem of elliptic equation by Levi function scheme. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Pan, Jinchao – PersonEntity: Name: NameFull: Liu, Jijun IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: Nov2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 0749159X Numbering: – Type: volume Value: 40 – Type: issue Value: 6 Titles: – TitleFull: Numerical Methods for Partial Differential Equations Type: main |
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