Convergence estimates for multigrid algorithms
To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the multigrid literature [1,2]. However, numerical evidence shows that Kaczmarz smoother does not satisfy above assumptions. Thus, we introduce a wea...
Saved in:
| Published in: | Computers & mathematics with applications (1987) Vol. 34; no. 9; pp. 15 - 22 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Ltd
01.11.1997
|
| Subjects: | |
| ISSN: | 0898-1221, 1873-7668 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the multigrid literature [1,2]. However, numerical evidence shows that Kaczmarz smoother does not satisfy above assumptions. Thus, we introduce a weaker condition which is satisfied by Kaczmarz smoother as well as Jacobi and Gauss-Seidel smoother. Under these weaker assumptions, we show that the convergence factor of
V-cycle multigrid algorithm is
δ =
1 − 1
(C(j − 1))
. assumptions for Kaczmarz smoother are verified by numerical experiment. |
|---|---|
| AbstractList | To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the multigrid literature [1,2]. However, numerical evidence shows that Kaczmarz smoother does not satisfy above assumptions. Thus, we introduce a weaker condition which is satisfied by Kaczmarz smoother as well as Jacobi and Gauss-Seidel smoother. Under these weaker assumptions, we show that the convergence factor of
V-cycle multigrid algorithm is
δ =
1 − 1
(C(j − 1))
. assumptions for Kaczmarz smoother are verified by numerical experiment. |
| Author | Kang, K.S. Kwak, D.Y. |
| Author_xml | – sequence: 1 givenname: K.S. surname: Kang fullname: Kang, K.S. email: kks002@math.kaist.ac.kr – sequence: 2 givenname: D.Y. surname: Kwak fullname: Kwak, D.Y. email: dykwak@math.kaist.ac.kr |
| BookMark | eNqFkE1LAzEURYNUsK3-BGGWupj6kkwmGVyIFL-g4EJdh0zypkamM5LEgv_emVZcuOnqweWeC-fNyKTrOyTknMKCAi2vXkBVKqeM0YtKXgJQJXJxRKZUSZ7LslQTMv2rnJBZjB8AUHAGU7JY9t0Wwxo7ixnG5DcmYcyaPmSbrzb5dfAuM-26Dz69b-IpOW5MG_Hs987J2_3d6_IxXz0_PC1vV7nlIFPuqGXM1Y7zhoEdMgqqhEJaZ4AXtVDCINSNFXWBTJaIaLgoKTYVp4VTnM_J9X7Xhj7GgI22Ppnk-y4F41tNQY_qeqeuRy9dSb1T12KgxT_6Mwxi4fsgd7PncFDbegw6Wj9-xvmANmnX-wMLP81rczg |
| CitedBy_id | crossref_primary_10_1016_j_apnum_2018_04_011 crossref_primary_10_1016_j_amc_2006_09_128 |
| Cites_doi | 10.1090/S0025-5718-1981-0595040-2 10.1090/S0025-5718-1992-1122058-0 10.1137/0720066 10.1137/0731089 10.1090/S0025-5718-1987-0906174-X 10.1090/S0025-5718-1991-1079008-4 10.1137/0722038 |
| ContentType | Journal Article |
| Copyright | 1997 |
| Copyright_xml | – notice: 1997 |
| DBID | 6I. AAFTH AAYXX CITATION |
| DOI | 10.1016/S0898-1221(97)00185-5 |
| DatabaseName | ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EISSN | 1873-7668 |
| EndPage | 22 |
| ExternalDocumentID | 10_1016_S0898_1221_97_00185_5 S0898122197001855 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 29F 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO AAYFN ABAOU ABBOA ABFNM ABMAC ABTAH ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACIWK ACNCT ACNNM ACRLP ACZNC ADBBV ADEZE ADMUD ADTZH AEBSH AECPX AEKER AENEX AEXQZ AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AHZHX AIALX AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CS3 DU5 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA GBOLZ HVGLF HZ~ IHE IXB J1W JJJVA KOM LG9 M26 M41 MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. PQQKQ Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSV SSW SSZ T5K TAE TN5 WUQ XFK XPP ZMT ZY4 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABJNI ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD |
| ID | FETCH-LOGICAL-c307t-d1c22dbd33f20cc301086047cda034b585ae0bfc5b4e276eeea3561ef9314d833 |
| ISICitedReferencesCount | 1 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=10_1016_S0898_1221_97_00185_5&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0898-1221 |
| IngestDate | Sat Nov 29 01:36:44 EST 2025 Tue Nov 18 22:07:16 EST 2025 Fri Feb 23 02:27:10 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 9 |
| Keywords | Multigrid method Smoothing assumptions Kaczmarz smoothing |
| Language | English |
| License | http://www.elsevier.com/open-access/userlicense/1.0 https://www.elsevier.com/tdm/userlicense/1.0 https://www.elsevier.com/open-access/userlicense/1.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c307t-d1c22dbd33f20cc301086047cda034b585ae0bfc5b4e276eeea3561ef9314d833 |
| OpenAccessLink | https://dx.doi.org/10.1016/S0898-1221(97)00185-5 |
| PageCount | 8 |
| ParticipantIDs | crossref_citationtrail_10_1016_S0898_1221_97_00185_5 crossref_primary_10_1016_S0898_1221_97_00185_5 elsevier_sciencedirect_doi_10_1016_S0898_1221_97_00185_5 |
| PublicationCentury | 1900 |
| PublicationDate | 1997-11-01 |
| PublicationDateYYYYMMDD | 1997-11-01 |
| PublicationDate_xml | – month: 11 year: 1997 text: 1997-11-01 day: 01 |
| PublicationDecade | 1990 |
| PublicationTitle | Computers & mathematics with applications (1987) |
| PublicationYear | 1997 |
| Publisher | Elsevier Ltd |
| Publisher_xml | – name: Elsevier Ltd |
| References | Bramble, Pasciak, Wang, Xu (BIB8) 1991; 56 Bramble, Pasciak (BIB7) 1993; 60 Bramble, Kwak, Pasciak (BIB6) 1994; 31 (BIB10) 1987 Bramble, Pasciak (BIB1) 1987; 49 Hackbush (BIB9) 1985 Bank, Dupont (BIB3) 1981; 36 Bramble, Pasciak (BIB2) 1992; 58 Braess, Hackbush (BIB5) 1983; 20 Bank, Douglas (BIB4) 1985; 22 Bank (10.1016/S0898-1221(97)00185-5_BIB4) 1985; 22 (10.1016/S0898-1221(97)00185-5_BIB10) 1987 Hackbush (10.1016/S0898-1221(97)00185-5_BIB9) 1985 Bramble (10.1016/S0898-1221(97)00185-5_BIB2) 1992; 58 Bramble (10.1016/S0898-1221(97)00185-5_BIB7) 1993; 60 Bramble (10.1016/S0898-1221(97)00185-5_BIB6) 1994; 31 Bramble (10.1016/S0898-1221(97)00185-5_BIB8) 1991; 56 Bramble (10.1016/S0898-1221(97)00185-5_BIB1) 1987; 49 Braess (10.1016/S0898-1221(97)00185-5_BIB5) 1983; 20 Bank (10.1016/S0898-1221(97)00185-5_BIB3) 1981; 36 |
| References_xml | – volume: 22 start-page: 617 year: 1985 end-page: 633 ident: BIB4 article-title: Sharp estimates for multigrid rates of convergence with general smoothing and acceleration publication-title: SIAM J. Numer. Anal. – volume: 31 start-page: 1746 year: 1994 end-page: 1763 ident: BIB6 article-title: Uniform convergence of multigrid publication-title: SIAM Numer. Anal. – year: 1985 ident: BIB9 article-title: Multi-Grid Methods and Applications – volume: 58 start-page: 467 year: 1992 end-page: 488 ident: BIB2 article-title: The analysis of smoothers for multigrid algorithms publication-title: Math. Comp. – volume: 49 start-page: 311 year: 1987 end-page: 329 ident: BIB1 article-title: New convergence estimates for multigrid algorithms publication-title: Math. Comp. – volume: 60 start-page: 447 year: 1993 end-page: 471 ident: BIB7 article-title: New estimates for multigrid algorithms including the publication-title: Math. Comp. – year: 1987 ident: BIB10 publication-title: Multigrid Methods – volume: 20 start-page: 967 year: 1983 end-page: 975 ident: BIB5 article-title: A new convergence proof for the multigrid method including the publication-title: SIAM J. Numer. Anal. – volume: 56 start-page: 23 year: 1991 end-page: 45 ident: BIB8 article-title: Convergence estimates for multigrid algorithms without regularity assumptions publication-title: Math. Comp. – volume: 36 start-page: 35 year: 1981 end-page: 51 ident: BIB3 article-title: An optimal order process for solving finite element equations publication-title: Math. Comp. – volume: 36 start-page: 35 year: 1981 ident: 10.1016/S0898-1221(97)00185-5_BIB3 article-title: An optimal order process for solving finite element equations publication-title: Math. Comp. doi: 10.1090/S0025-5718-1981-0595040-2 – year: 1987 ident: 10.1016/S0898-1221(97)00185-5_BIB10 – year: 1985 ident: 10.1016/S0898-1221(97)00185-5_BIB9 – volume: 58 start-page: 467 year: 1992 ident: 10.1016/S0898-1221(97)00185-5_BIB2 article-title: The analysis of smoothers for multigrid algorithms publication-title: Math. Comp. doi: 10.1090/S0025-5718-1992-1122058-0 – volume: 20 start-page: 967 year: 1983 ident: 10.1016/S0898-1221(97)00185-5_BIB5 article-title: A new convergence proof for the multigrid method including the V-cycle publication-title: SIAM J. Numer. Anal. doi: 10.1137/0720066 – volume: 31 start-page: 1746 year: 1994 ident: 10.1016/S0898-1221(97)00185-5_BIB6 article-title: Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems publication-title: SIAM Numer. Anal. doi: 10.1137/0731089 – volume: 49 start-page: 311 year: 1987 ident: 10.1016/S0898-1221(97)00185-5_BIB1 article-title: New convergence estimates for multigrid algorithms publication-title: Math. Comp. doi: 10.1090/S0025-5718-1987-0906174-X – volume: 56 start-page: 23 year: 1991 ident: 10.1016/S0898-1221(97)00185-5_BIB8 article-title: Convergence estimates for multigrid algorithms without regularity assumptions publication-title: Math. Comp. doi: 10.1090/S0025-5718-1991-1079008-4 – volume: 22 start-page: 617 year: 1985 ident: 10.1016/S0898-1221(97)00185-5_BIB4 article-title: Sharp estimates for multigrid rates of convergence with general smoothing and acceleration publication-title: SIAM J. Numer. Anal. doi: 10.1137/0722038 – volume: 60 start-page: 447 year: 1993 ident: 10.1016/S0898-1221(97)00185-5_BIB7 article-title: New estimates for multigrid algorithms including the V-cycle publication-title: Math. Comp. |
| SSID | ssj0004320 |
| Score | 1.5129596 |
| Snippet | To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 15 |
| SubjectTerms | Kaczmarz smoothing Multigrid method Smoothing assumptions |
| Title | Convergence estimates for multigrid algorithms |
| URI | https://dx.doi.org/10.1016/S0898-1221(97)00185-5 |
| Volume | 34 |
| WOSCitedRecordID | wos10_1016_S0898_1221_97_00185_5&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1873-7668 dateEnd: 20180131 omitProxy: false ssIdentifier: ssj0004320 issn: 0898-1221 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT9wwELbKwoFLX1CVvpQDh1bIaRLHcXxcbanaUqFKUGk5RYljLytBQNkt8PM7fsRe1Io-JC5RZPmheD6Nx5OZbxDarUH1taoosCpIgrVFjJtcCZwWVJBGybxoDGX-V3Z4WE6n_JsLa16YcgKs68qbG355r6KGNhC2Tp39B3H7SaEB3kHo8ASxw_OvBD_RceS95djUFBrn2po00YQmeHDWz9u9-mx20c-Xp46qfGAqcBUeFgYP557QdciAW_nXbfidbjsSDpzn-SA-in3bdW207Yf4JA7-BcPNmt7yL_jElxBlZHQTh8tnZpObY2l1Z8kIZoWtkjMoV-eptCDiK5rSJnG6M9emJv-iza1j4cgvBp-meYO5riRIMQ1HmA8sNH11V85MJ7qG1jNGeTlC6-PP-9MvIWeWWMrOYe6Q3fU-LPiWs3dusd_bLSu2yPFj9NBdIqKxFf4T9EB2T9GjQXyR09dbKF7BQuSxEAEWIo-FKGBhG33_uH88-YRdgQwsQDUvcZuKLGublhCVJQLadNmsJGeirROSN3ATrGXSKEGbXGaskFLWBOxlqThJ87Yk5BkadRedfI4i2qqSU5hJW6SKixIOnjqVXGawBmVkB-XD51fCscfrIiZnVQgThF2r9K5VJqYBdq2iOyj2wy4tfcqfBpTD3lbOBrS2XQWguHvoi_8f-hJtBui_QqNl_0O-Rhviajlf9G8cdH4CceZ3XA |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Convergence+estimates+for+multigrid+algorithms&rft.jtitle=Computers+%26+mathematics+with+applications+%281987%29&rft.au=Kang%2C+K.S.&rft.au=Kwak%2C+D.Y.&rft.date=1997-11-01&rft.pub=Elsevier+Ltd&rft.issn=0898-1221&rft.eissn=1873-7668&rft.volume=34&rft.issue=9&rft.spage=15&rft.epage=22&rft_id=info:doi/10.1016%2FS0898-1221%2897%2900185-5&rft.externalDocID=S0898122197001855 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0898-1221&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0898-1221&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0898-1221&client=summon |