On the convergence in H1-norm for the fractional Laplacian.
Resumen:
We consider the numerical solution of the fractional Laplacian of index s∈(1/2,1) in a bounded domain Ω with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space H˜s(Ω). For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in H1(Ω). In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to H1(Ω). A natural question is then whether one can obtain error estimates in H1(Ω)-norm, in addition to the classical ones that can be derived in the H˜s(Ω) energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.
| 2018 | |
| Juan Pablo Borthagaray ha sido financiado en parte por la subvención DMS-1411808 de la NSF. | |
|
Fractional Laplacian Finite elements Graded meshes |
|
| Inglés | |
| Universidad de la República | |
| COLIBRI | |
| https://hdl.handle.net/20.500.12008/47775 | |
| Acceso abierto | |
| Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| _version_ | 1875693216768983040 |
|---|---|
| author | Borthagaray, Juan Pablo |
| author2 | Ciarlet Jr, Patrick |
| author2_role | author |
| author_facet | Borthagaray, Juan Pablo Ciarlet Jr, Patrick |
| author_role | author |
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| collection | COLIBRI |
| dc.contributor.filiacion.none.fl_str_mv | Borthagaray Juan Pablo, Universidad de la República (Uruguay). Facultad de Ingeniería. Ciarlet Jr Patrick, Université Paris-Saclay, France |
| dc.creator.none.fl_str_mv | Borthagaray, Juan Pablo Ciarlet Jr, Patrick |
| dc.date.accessioned.none.fl_str_mv | 2024-12-26T15:57:40Z |
| dc.date.available.none.fl_str_mv | 2024-12-26T15:57:40Z |
| dc.date.issued.none.fl_str_mv | 2018 |
| dc.description.abstract.none.fl_txt_mv | We consider the numerical solution of the fractional Laplacian of index s∈(1/2,1) in a bounded domain Ω with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space H˜s(Ω). For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in H1(Ω). In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to H1(Ω). A natural question is then whether one can obtain error estimates in H1(Ω)-norm, in addition to the classical ones that can be derived in the H˜s(Ω) energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes. |
| dc.description.es.fl_txt_mv | También publicado en SIAM Journal on Numerical Analysis, vol. 57, no 4, 2019, pp. 1723-1743. DOI : 10.1137/18M122143 |
| dc.description.sponsorship.none.fl_txt_mv | Juan Pablo Borthagaray ha sido financiado en parte por la subvención DMS-1411808 de la NSF. |
| dc.format.extent.es.fl_str_mv | 19 p. |
| dc.format.mimetype.es.fl_str_mv | application/pdf |
| dc.identifier.citation.es.fl_str_mv | Borthagaray, J y Ciarlet Jr, P. On the convergence in H1-norm for the fractional Laplacian. [Preprint]. Publicado en: Mathematics. Numerical Analysis (math.NA), 2018, pp 1-19. arXiv:1810.07645v1. |
| dc.identifier.uri.none.fl_str_mv | https://hdl.handle.net/20.500.12008/47775 |
| dc.language.iso.none.fl_str_mv | en eng |
| dc.publisher.es.fl_str_mv | arXiv |
| dc.relation.none.fl_str_mv | Mathematics. Numerical Analysis (math.NA), arXiv:1810.07645v1, oct. 2018, pp. 1-19. |
| dc.rights.license.none.fl_str_mv | Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| dc.rights.none.fl_str_mv | info:eu-repo/semantics/openAccess |
| dc.source.none.fl_str_mv | reponame:COLIBRI instname:Universidad de la República instacron:Universidad de la República |
| dc.subject.es.fl_str_mv | Fractional Laplacian Finite elements Graded meshes |
| dc.title.none.fl_str_mv | On the convergence in H1-norm for the fractional Laplacian. |
| dc.type.es.fl_str_mv | Preprint |
| dc.type.none.fl_str_mv | info:eu-repo/semantics/preprint |
| dc.type.version.none.fl_str_mv | info:eu-repo/semantics/submittedVersion |
| description | También publicado en SIAM Journal on Numerical Analysis, vol. 57, no 4, 2019, pp. 1723-1743. DOI : 10.1137/18M122143 |
| eu_rights_str_mv | openAccess |
| format | preprint |
| id | COLIBRI_9fa06ccea0b04b6fb48528175eeb6812 |
| identifier_str_mv | Borthagaray, J y Ciarlet Jr, P. On the convergence in H1-norm for the fractional Laplacian. [Preprint]. Publicado en: Mathematics. Numerical Analysis (math.NA), 2018, pp 1-19. arXiv:1810.07645v1. |
| instacron_str | Universidad de la República |
| institution | Universidad de la República |
| instname_str | Universidad de la República |
| language | eng |
| language_invalid_str_mv | en |
| network_acronym_str | COLIBRI |
| network_name_str | COLIBRI |
| oai_identifier_str | oai:colibri.udelar.edu.uy:20.500.12008/47775 |
| publishDate | 2018 |
| reponame_str | COLIBRI |
| repository.mail.fl_str_mv | karina.camps@seciu.edu.uy |
| repository.name.fl_str_mv | COLIBRI - Universidad de la República |
| repository_id_str | 4771 |
| rights_invalid_str_mv | Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| spelling | Borthagaray Juan Pablo, Universidad de la República (Uruguay). Facultad de Ingeniería.Ciarlet Jr Patrick, Université Paris-Saclay, France2024-12-26T15:57:40Z2024-12-26T15:57:40Z2018Borthagaray, J y Ciarlet Jr, P. On the convergence in H1-norm for the fractional Laplacian. [Preprint]. Publicado en: Mathematics. Numerical Analysis (math.NA), 2018, pp 1-19. arXiv:1810.07645v1.https://hdl.handle.net/20.500.12008/47775También publicado en SIAM Journal on Numerical Analysis, vol. 57, no 4, 2019, pp. 1723-1743. DOI : 10.1137/18M122143We consider the numerical solution of the fractional Laplacian of index s∈(1/2,1) in a bounded domain Ω with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space H˜s(Ω). For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in H1(Ω). In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to H1(Ω). A natural question is then whether one can obtain error estimates in H1(Ω)-norm, in addition to the classical ones that can be derived in the H˜s(Ω) energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.Submitted by Ribeiro Jorge (jribeiro@fing.edu.uy) on 2024-12-19T18:34:19Z No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 13adb202270a5f7cee03e795b33133c4 (MD5) BC18.pdf: 573382 bytes, checksum: d6d723e9ca42fd12fde575cdb7ec76ac (MD5)Approved for entry into archive by Machado Jimena (jmachado@fing.edu.uy) on 2024-12-26T15:01:42Z (GMT) No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 13adb202270a5f7cee03e795b33133c4 (MD5) BC18.pdf: 573382 bytes, checksum: d6d723e9ca42fd12fde575cdb7ec76ac (MD5)Made available in DSpace by Luna Fabiana (fabiana.luna@seciu.edu.uy) on 2024-12-26T15:57:40Z (GMT). No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 13adb202270a5f7cee03e795b33133c4 (MD5) BC18.pdf: 573382 bytes, checksum: d6d723e9ca42fd12fde575cdb7ec76ac (MD5) Previous issue date: 2018Juan Pablo Borthagaray ha sido financiado en parte por la subvención DMS-1411808 de la NSF.19 p.application/pdfenengarXivMathematics. Numerical Analysis (math.NA), arXiv:1810.07645v1, oct. 2018, pp. 1-19.Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)Fractional LaplacianFinite elementsGraded meshesOn the convergence in H1-norm for the fractional Laplacian.Preprintinfo:eu-repo/semantics/preprintinfo:eu-repo/semantics/submittedVersionreponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaBorthagaray, Juan PabloCiarlet Jr, PatrickLICENSElicense.txtlicense.txttext/plain; charset=utf-84267http://localhost:8080/xmlui/bitstream/20.500.12008/47775/5/license.txt6429389a7df7277b72b7924fdc7d47a9MD55CC-LICENSElicense_urllicense_urltext/plain; charset=utf-850http://localhost:8080/xmlui/bitstream/20.500.12008/47775/2/license_urla006180e3f5b2ad0b88185d14284c0e0MD52license_textlicense_texttext/html; charset=utf-829882http://localhost:8080/xmlui/bitstream/20.500.12008/47775/3/license_text5499c15bd6b32c37160bfba5f71ab0a0MD53license_rdflicense_rdfapplication/rdf+xml; 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- Universidad de la Repúblicafalse |
| spellingShingle | On the convergence in H1-norm for the fractional Laplacian. Borthagaray, Juan Pablo Fractional Laplacian Finite elements Graded meshes |
| status_str | submittedVersion |
| title | On the convergence in H1-norm for the fractional Laplacian. |
| title_full | On the convergence in H1-norm for the fractional Laplacian. |
| title_fullStr | On the convergence in H1-norm for the fractional Laplacian. |
| title_full_unstemmed | On the convergence in H1-norm for the fractional Laplacian. |
| title_short | On the convergence in H1-norm for the fractional Laplacian. |
| title_sort | On the convergence in H1-norm for the fractional Laplacian. |
| topic | Fractional Laplacian Finite elements Graded meshes |
| url | https://hdl.handle.net/20.500.12008/47775 |
