SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS
This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the it...
Gespeichert in:
| Veröffentlicht in: | SIAM journal on numerical analysis 2009-01, Vol.47 (1), p.646-674 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 674 |
|---|---|
| container_issue | 1 |
| container_start_page | 646 |
| container_title | SIAM journal on numerical analysis |
| container_volume | 47 |
| creator | AMOSOV, ANDREY AHUES, MARIO LARGILLIER, ALAIN |
| description | This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme. |
| doi_str_mv | 10.1137/070685464 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_hal_p</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_00866839v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>25663140</jstor_id><sourcerecordid>25663140</sourcerecordid><originalsourceid>FETCH-LOGICAL-c312t-47b28f5d4c1162da159bc5f6f359e7cfee58e6091ce3aa94e0e1036aca07ddfe3</originalsourceid><addsrcrecordid>eNpFkE9Pg0AQxTdGE2v14Acw2XjzgO6wsMCR4JaiFCoU_5w2W1him1YqtCZ-e5e0qafJe_PLy8xD6BrIPQB1HohDmGtbzDpBAyCebTjgkFM0IIQyAyzTO0cXXbckWrtAB2idF1OeBWnyyrOQJwHH6Qjn6YTjaZY-8WAWpQn2p1q8RxO_VzkepRl-4_5z_IHzKAmL2M9wlMx4mPkx5i_FASv6JdahvPfDLHrML9FZLVedujrMISpGfBaMjTgNo8CPjZKCuTUsZ266tV1ZJQAzKwm2Ny_tmtXU9pRT1krZrmLEg1JRKT1LEQX6P1lK4lRVregQ3e1zP-VKbNrFWra_opELMfZj0XuEuIy51PsBzd7u2U3bfO9UtxXLZtd-6fOEZ5rAqEb_A8u26bpW1cdUIKIvXhyL1-zNnl1226Y9gqbNGAWL0D-6VXTg</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>922163086</pqid></control><display><type>article</type><title>SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS</title><source>SIAM Journals Online</source><source>JSTOR Mathematics & Statistics</source><source>Jstor Complete Legacy</source><creator>AMOSOV, ANDREY ; AHUES, MARIO ; LARGILLIER, ALAIN</creator><creatorcontrib>AMOSOV, ANDREY ; AHUES, MARIO ; LARGILLIER, ALAIN</creatorcontrib><description>This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme.</description><identifier>ISSN: 0036-1429</identifier><identifier>EISSN: 1095-7170</identifier><identifier>DOI: 10.1137/070685464</identifier><language>eng</language><publisher>Philadelphia: Society for Industrial and Applied Mathematics</publisher><subject>Applied mathematics ; Approximation ; Astrophysics ; Banach spaces ; Differential equations ; Error rates ; Galerkin methods ; Integral equations ; Mathematical constants ; Mathematical functions ; Mathematics ; Numerical Analysis ; Singular integral equations</subject><ispartof>SIAM journal on numerical analysis, 2009-01, Vol.47 (1), p.646-674</ispartof><rights>Copyright ©2010 Society for Industrial and Applied Mathematics</rights><rights>[Copyright] © 2009 Society for Industrial and Applied Mathematics</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c312t-47b28f5d4c1162da159bc5f6f359e7cfee58e6091ce3aa94e0e1036aca07ddfe3</citedby><cites>FETCH-LOGICAL-c312t-47b28f5d4c1162da159bc5f6f359e7cfee58e6091ce3aa94e0e1036aca07ddfe3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/25663140$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/25663140$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,3170,27903,27904,57995,57999,58228,58232</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00866839$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>AMOSOV, ANDREY</creatorcontrib><creatorcontrib>AHUES, MARIO</creatorcontrib><creatorcontrib>LARGILLIER, ALAIN</creatorcontrib><title>SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS</title><title>SIAM journal on numerical analysis</title><description>This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme.</description><subject>Applied mathematics</subject><subject>Approximation</subject><subject>Astrophysics</subject><subject>Banach spaces</subject><subject>Differential equations</subject><subject>Error rates</subject><subject>Galerkin methods</subject><subject>Integral equations</subject><subject>Mathematical constants</subject><subject>Mathematical functions</subject><subject>Mathematics</subject><subject>Numerical Analysis</subject><subject>Singular integral equations</subject><issn>0036-1429</issn><issn>1095-7170</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNpFkE9Pg0AQxTdGE2v14Acw2XjzgO6wsMCR4JaiFCoU_5w2W1him1YqtCZ-e5e0qafJe_PLy8xD6BrIPQB1HohDmGtbzDpBAyCebTjgkFM0IIQyAyzTO0cXXbckWrtAB2idF1OeBWnyyrOQJwHH6Qjn6YTjaZY-8WAWpQn2p1q8RxO_VzkepRl-4_5z_IHzKAmL2M9wlMx4mPkx5i_FASv6JdahvPfDLHrML9FZLVedujrMISpGfBaMjTgNo8CPjZKCuTUsZ266tV1ZJQAzKwm2Ny_tmtXU9pRT1krZrmLEg1JRKT1LEQX6P1lK4lRVregQ3e1zP-VKbNrFWra_opELMfZj0XuEuIy51PsBzd7u2U3bfO9UtxXLZtd-6fOEZ5rAqEb_A8u26bpW1cdUIKIvXhyL1-zNnl1226Y9gqbNGAWL0D-6VXTg</recordid><startdate>20090101</startdate><enddate>20090101</enddate><creator>AMOSOV, ANDREY</creator><creator>AHUES, MARIO</creator><creator>LARGILLIER, ALAIN</creator><general>Society for Industrial and Applied Mathematics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7WY</scope><scope>7WZ</scope><scope>7X2</scope><scope>7XB</scope><scope>87Z</scope><scope>88A</scope><scope>88F</scope><scope>88I</scope><scope>88K</scope><scope>8AL</scope><scope>8FE</scope><scope>8FG</scope><scope>8FH</scope><scope>8FK</scope><scope>8FL</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>ATCPS</scope><scope>AZQEC</scope><scope>BBNVY</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>BHPHI</scope><scope>CCPQU</scope><scope>D1I</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>KB.</scope><scope>L.-</scope><scope>L6V</scope><scope>LK8</scope><scope>M0C</scope><scope>M0K</scope><scope>M0N</scope><scope>M1Q</scope><scope>M2O</scope><scope>M2P</scope><scope>M2T</scope><scope>M7P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PATMY</scope><scope>PDBOC</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>PYCSY</scope><scope>Q9U</scope><scope>1XC</scope></search><sort><creationdate>20090101</creationdate><title>SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS</title><author>AMOSOV, ANDREY ; AHUES, MARIO ; LARGILLIER, ALAIN</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c312t-47b28f5d4c1162da159bc5f6f359e7cfee58e6091ce3aa94e0e1036aca07ddfe3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Applied mathematics</topic><topic>Approximation</topic><topic>Astrophysics</topic><topic>Banach spaces</topic><topic>Differential equations</topic><topic>Error rates</topic><topic>Galerkin methods</topic><topic>Integral equations</topic><topic>Mathematical constants</topic><topic>Mathematical functions</topic><topic>Mathematics</topic><topic>Numerical Analysis</topic><topic>Singular integral equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>AMOSOV, ANDREY</creatorcontrib><creatorcontrib>AHUES, MARIO</creatorcontrib><creatorcontrib>LARGILLIER, ALAIN</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>Agricultural Science Collection</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Biology Database (Alumni Edition)</collection><collection>Military Database (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Telecommunications (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Natural Science Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>Agricultural & Environmental Science Collection</collection><collection>ProQuest Central Essentials</collection><collection>Biological Science Collection</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>Natural Science Collection (ProQuest)</collection><collection>ProQuest One Community College</collection><collection>ProQuest Materials Science Collection</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>Materials Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>ProQuest Biological Science Collection</collection><collection>ABI/INFORM Global</collection><collection>Agricultural Science Database</collection><collection>Computing Database</collection><collection>Military Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Telecommunications Database</collection><collection>Biological Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Environmental Science Database</collection><collection>Materials Science Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>Environmental Science Collection</collection><collection>ProQuest Central Basic</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>SIAM journal on numerical analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>AMOSOV, ANDREY</au><au>AHUES, MARIO</au><au>LARGILLIER, ALAIN</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>2009-01-01</date><risdate>2009</risdate><volume>47</volume><issue>1</issue><spage>646</spage><epage>674</epage><pages>646-674</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><abstract>This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme.</abstract><cop>Philadelphia</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/070685464</doi><tpages>29</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0036-1429 |
| ispartof | SIAM journal on numerical analysis, 2009-01, Vol.47 (1), p.646-674 |
| issn | 0036-1429 1095-7170 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00866839v1 |
| source | SIAM Journals Online; JSTOR Mathematics & Statistics; Jstor Complete Legacy |
| subjects | Applied mathematics Approximation Astrophysics Banach spaces Differential equations Error rates Galerkin methods Integral equations Mathematical constants Mathematical functions Mathematics Numerical Analysis Singular integral equations |
| title | SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-24T12%3A10%3A24IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_hal_p&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=SUPERCONVERGENCE%20OF%20SOME%20PROJECTION%20APPROXIMATIONS%20FOR%20WEAKLY%20SINGULAR%20INTEGRAL%20EQUATIONS%20USING%20GENERAL%20GRIDS&rft.jtitle=SIAM%20journal%20on%20numerical%20analysis&rft.au=AMOSOV,%20ANDREY&rft.date=2009-01-01&rft.volume=47&rft.issue=1&rft.spage=646&rft.epage=674&rft.pages=646-674&rft.issn=0036-1429&rft.eissn=1095-7170&rft_id=info:doi/10.1137/070685464&rft_dat=%3Cjstor_hal_p%3E25663140%3C/jstor_hal_p%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=922163086&rft_id=info:pmid/&rft_jstor_id=25663140&rfr_iscdi=true |