Algebraic multigrid methods based on element preconditioning
This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/St...
Saved in:
| Published in | International journal of computer mathematics Vol. 78; no. 4; pp. 575 - 598 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Gordon and Breach Science Publishers
01.01.2001
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0020-7160 1029-0265 |
| DOI | 10.1080/00207160108805133 |
Cover
| Abstract | This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the "region of robustness" between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.
For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.
The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements. |
|---|---|
| AbstractList | This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the "region of robustness" between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.
For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.
The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements. |
| Author | Reitzinger, S. Schöberl, J. Haase, G Langer, U. |
| Author_xml | – sequence: 1 givenname: G surname: Haase fullname: Haase, G organization: Department of Computational Mathematics and Optimization , Institute for Analysis and Computational Mathematics – sequence: 2 givenname: U. surname: Langer fullname: Langer, U. email: ulanger@numa.uni-linz.ac.at organization: Department of Computational Mathematics and Optimization , Institute for Analysis and Computational Mathematics – sequence: 3 givenname: S. surname: Reitzinger fullname: Reitzinger, S. organization: Special Research Program , Numerical and Symbolic Scientific Computing – sequence: 4 givenname: J. surname: Schöberl fullname: Schöberl, J. organization: Special Research Program , Numerical and Symbolic Scientific Computing |
| BookMark | eNqFkM1KAzEUhYMo2FYfwN28wOhNMp1JoZtS_IOCm-6HTHJTI5mkJBHt2zuDXVnU1eX-fPcczpSc--CRkBsKtxQE3AEwaGgNQyNgTjk_IxMKbFECq-fnZDLuy_HgkkxTegMAsWjqCVmu3A67KK0q-neX7S5aXfSYX4NORScT6iL4Ah326HOxj6iC1zbb4K3fXZELI13C62Odke3D_Xb9VG5eHp_Xq02peFXlUhrKgAM1ogLGG8EM7YTgWkvB62HIEA1XlCmlaINzxSrd1ZTXiB0VleAz0ny_VTGkFNG0ymY5esiDcddSaMcM2pMMBpL-IPfR9jIe_mSOatabEHv5EaLTbZYHF6KJ0iubTqk2f-aBXP5L8t-FvwDAVYRy |
| CitedBy_id | crossref_primary_10_1142_S0218202503002519 crossref_primary_10_1007_s00607_005_0147_x crossref_primary_10_1137_050641156 crossref_primary_10_1137_S1064827501386237 crossref_primary_10_1080_17445760601122084 crossref_primary_10_1137_060675940 crossref_primary_10_1002_nla_2408 crossref_primary_10_1002_nme_2853 crossref_primary_10_1016_j_pbiomolbio_2007_07_012 crossref_primary_10_1137_S1064827502405112 crossref_primary_10_1137_070679673 crossref_primary_10_1016_j_advengsoft_2006_08_006 |
| Cites_doi | 10.1007/978-3-663-01354-9 10.1007/978-3-642-58734-4_9 10.1007/BF02238511 10.1007/978-3-662-02427-0 10.1016/B978-008043568-8/50015-8 10.1080/00207169208804106 10.1007/BF01395810 10.1007/978-3-642-58312-4_17 10.1016/0096-3003(86)90095-0 10.1007/BF02238488 |
| ContentType | Journal Article |
| Copyright | Copyright Taylor & Francis Group, LLC 2001 |
| Copyright_xml | – notice: Copyright Taylor & Francis Group, LLC 2001 |
| DBID | AAYXX CITATION |
| DOI | 10.1080/00207160108805133 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EISSN | 1029-0265 |
| EndPage | 598 |
| ExternalDocumentID | 10_1080_00207160108805133 8805133 |
| GroupedDBID | -~X .7F .QJ 0BK 0R~ 1TA 30N 4.4 5GY 5VS AAENE AAJMT AALDU AAMIU AAPUL AAQRR ABCCY ABFIM ABHAV ABJNI ABLIJ ABPAQ ABPEM ABTAI ABXUL ABXYU ACAGQ ACGEJ ACGFS ACGOD ACIWK ACNCT ADCVX ADGTB ADXPE AEISY AENEX AEOZL AEPSL AEYOC AFKVX AGDLA AGMYJ AGROQ AHDZW AHMOU AI. AIJEM AJWEG AKBVH AKOOK ALCKM ALMA_UNASSIGNED_HOLDINGS ALQZU AMEWO AVBZW AWYRJ BLEHA CAG CCCUG CE4 COF CRFIH CS3 DGEBU DKSSO DMQIW DU5 EBS EJD ESX E~A E~B GTTXZ H13 HZ~ H~P J.P KYCEM M4Z NA5 NY~ O9- P2P PQQKQ QCRFL RNANH ROSJB RTWRZ S-T SNACF TBQAZ TDBHL TEJ TFL TFT TFW TN5 TOXWX TTHFI TUROJ TWF UPT UT5 UU3 VH1 WH7 ZGOLN ~S~ .4S .DC 07G 29J AAGDL AAHIA AAIKQ AAKBW AAYJJ AAYXX ABDBF ABEFU ACGEE ACTCW ACTIO ACUHS AEUMN AFFNX AFRVT AGCQS AGLEN AIYEW AMVHM AMXXU AQRUH ARCSS BCCOT BPLKW C06 CITATION DWIFK EAP EDO EMK EPL EST HF~ H~9 IPNFZ IVXBP LJTGL MK~ NUSFT RIG TAQ TASJS TFMCV TUS UB9 UU8 V3K V4Q ZY4 |
| ID | FETCH-LOGICAL-c344t-af120301f84023782f1b883dda836f842eef3c12ccc17e5c24db6136eeb18483 |
| ISSN | 0020-7160 |
| IngestDate | Thu Apr 24 23:02:55 EDT 2025 Wed Oct 01 01:48:27 EDT 2025 Wed Dec 25 09:00:53 EST 2024 Mon May 13 12:10:04 EDT 2019 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c344t-af120301f84023782f1b883dda836f842eef3c12ccc17e5c24db6136eeb18483 |
| PageCount | 24 |
| ParticipantIDs | crossref_primary_10_1080_00207160108805133 crossref_citationtrail_10_1080_00207160108805133 informaworld_taylorfrancis_310_1080_00207160108805133 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 1/1/2001 2001-01-00 |
| PublicationDateYYYYMMDD | 2001-01-01 |
| PublicationDate_xml | – month: 01 year: 2001 text: 1/1/2001 day: 01 |
| PublicationDecade | 2000 |
| PublicationTitle | International journal of computer mathematics |
| PublicationYear | 2001 |
| Publisher | Gordon and Breach Science Publishers |
| Publisher_xml | – name: Gordon and Breach Science Publishers |
| References | Hackbusch W. (CIT0009) 1985 CIT0012 CIT0023 Brandt A. (CIT0005) 1984 CIT0022 Jung M. (CIT0011) 1991; 1 Ruge J.W. (CIT0020) 1985; 3 Hackbusch W. (CIT0010) 1991 Grauschopf T. (CIT0006) 1996 Haase G. (CIT0007) 2000; 10 Bank R.E. (CIT0001) 1980 Kaltenbacher M. (CIT0014) 2000 CIT0003 Brandt A. (CIT0004) 1982 Meurant G. (CIT0018) 1990; 28 Ruge J.W. (CIT0021) 1986; 5 CIT0002 CIT0013 CIT0016 CIT0015 CIT0008 |
| References_xml | – volume: 5 start-page: 73 volume-title: Frontiers in Applied Mathematics year: 1986 ident: CIT0021 – volume-title: Iterative Lösung groβer schwachbesetzter Gleichungssysteme year: 1991 ident: CIT0010 doi: 10.1007/978-3-663-01354-9 – ident: CIT0015 doi: 10.1007/978-3-642-58734-4_9 – ident: CIT0023 doi: 10.1007/BF02238511 – volume-title: Algebraic Multigrid (AMG) for automatic multigrid solution with application to geodetic computations year: 1982 ident: CIT0004 – volume-title: Multigrid methods and application year: 1985 ident: CIT0009 doi: 10.1007/978-3-662-02427-0 – ident: CIT0012 – ident: CIT0016 doi: 10.1016/B978-008043568-8/50015-8 – volume: 28 volume-title: Studies in Mathematics and its Applications year: 1990 ident: CIT0018 – year: 2000 ident: CIT0014 publication-title: Algebraic multigrid for static nonlinear 3D electromagnetic field compuations – start-page: 257 volume-title: Sparsity and its Application year: 1984 ident: CIT0005 – volume: 3 start-page: 169 volume-title: Multigrid Methods for integral and differential equations year: 1985 ident: CIT0020 – ident: CIT0008 doi: 10.1080/00207169208804106 – volume: 10 start-page: 41 year: 2000 ident: CIT0007 publication-title: Electronic Transactions on Numerical Analysis {ETNA) – ident: CIT0022 doi: 10.1007/BF01395810 – ident: CIT0013 doi: 10.1007/978-3-642-58312-4_17 – ident: CIT0003 doi: 10.1016/0096-3003(86)90095-0 – volume: 1 start-page: 217 year: 1991 ident: CIT0011 publication-title: Surveys Math. Indust – ident: CIT0002 doi: 10.1007/BF02238488 – volume-title: Analysis of a two-level scheme for solving finite element equations year: 1980 ident: CIT0001 – volume-title: Additive Multilevel Preconditioners based on Bilinear Interpolation, matrix dependent geometric coarsening and algebraic multigrid coarsening for second order elliptic PDEs year: 1996 ident: CIT0006 |
| SSID | ssj0008976 |
| Score | 1.5990329 |
| Snippet | This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element... |
| SourceID | crossref informaworld |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 575 |
| SubjectTerms | Algebraic Multigrid Element Preconditioning Technique Finite Element Equations G1.3 G1.8 Preconditioned Conjugate Gradient Method |
| Title | Algebraic multigrid methods based on element preconditioning |
| URI | https://www.tandfonline.com/doi/abs/10.1080/00207160108805133 |
| Volume | 78 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVLSH databaseName: aylor and Francis Online customDbUrl: mediaType: online eissn: 1029-0265 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0008976 issn: 0020-7160 databaseCode: AHDZW dateStart: 19970101 isFulltext: true providerName: Library Specific Holdings – providerCode: PRVAWR databaseName: Taylor & Francis Science and Technology Library-DRAA customDbUrl: eissn: 1029-0265 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0008976 issn: 0020-7160 databaseCode: 30N dateStart: 19970101 isFulltext: true titleUrlDefault: http://www.tandfonline.com/page/title-lists providerName: Taylor & Francis |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3NT9swFLdKuezCgA3RDZAPO4FSJXacuhKXwj6qCbhQNMSlsh2nRIKCSiYhbvznPNeO6xaGBpeofY2Tyu_Z79Pvh9A3UNoyJxzcVKFZlHIpo66gnchY35opJrQ0Gd3jk6x_lv4-Z-eNxmN4uqSSbfXw4rmS93AVaMBXc0r2DZz1DwUCfAb-whU4DNf_4nHvamTyvqWrCxxNytxBQt_tGfWUm1SAtgXiphsA-L556SKwoVU6HxYMmkkoh_mwd-27u97N9ixhQRk9OtfR9AyxIZ21fSJHl9VDWdNPPf1UXZoc_UEm9cRGodtzAYhkIQDxC5xkVzh9MDEVoH5XmhX2hzswuKvgo9lcjLabrinBAV-QhbtyhwfSlwZbLLNIK05bM4th_UwR1JWTJDYvg688Nkg2M63naxHdL0tomXSyjDTRcq___eKPV-e8O0Uo9H-8To2bBu2Lj58zbuZa3wZGy2AVrThvA_es6Kyhhh6vo481kgd2U_gJ7XtJwl6SsJMkPJUkfDPGTpLwgiR9RoOfPwaH_cjhakSKpmkViSIhZi0W4NwTCiZikUjOaZ4LTjMgEq0LqhKilEo6sGBJmkuw-jINep2nnG6g5vhmrDcRllxmipJCSQnDRNZVIqZCx0yAVZ2IvIXiej6GyvWcN9AnV8PEt6ZdmMIW2vVDbm3DlddujsNJHlbTKFdhIWme3z6s7qsWYq8Mof981Zd3jvuKPswWzRZqVpO_ehvs2EruOEl7AhgfmDU |
| linkProvider | Library Specific Holdings |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Ja8JAFH609tBealdq1zn0VIgkmSSO0IuUim3VkwVvYbaIVFQ0Qumv77xMImqXg5BTeC_LbG-ZN98HcG-MtlA-M2Eq16ETMCGcOqc1B71vHcqQa4E7up1u1HoPXvthP0-4zfOySoyhEwsUka3VOLkxGV2UxOERbmMZMZQwow8JSnZhL0SUETzC4XaXKzGrZ-RyKO6gfLGr-dsj1uzSGmrpir1pliEuvtSWmXxUF6moyq8NEMftf-UIDnNXlDTs2DmGHT0-gXJB80DyWX8Kj43RALeXh5Jk5YeD2VARyzw9J2gFFZmMibZ16GSahdhqmCd6z6DXfO49tZycdMGRNAhShyeejx2VmMjPp8Z_SDzBGFWKMxqZm77WCZWeL6X0aqY3_UAJ4xJE2iz6LGD0HErjyVhfABFMRJL6iRTCqPGoLrlLuXZDblwuj6sKuEWLxzIHJEdejFHsLXFLNxqnAg9LlalF4_hP2F3txjjNUiCJ5Sv5KR6nn2kFwn9U6J-vutxS7w72W71OO26_dN-u4MDWt-F1DaV0ttA3xuFJxW02qr8BoQfyPw |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1La8JAEB5aC6WX2ie1zz30VIgk2SSu0Iu0FfuSHix4C_sUqUTRCKW_vjtJlGpbD0JOYSYhu5udmd1vvw_g2gZtoXxmy1SuQydgQjh1TmsOZt86lCHXAnd0X9tR6z146obdApszKWCVWEObnCgim6vx5x4pM0PE4QluGxixkrCDD_VJNmErwjOmeILDbc8nYlbPtOXQ3EH72abmX49YCEsLpKU_wk2znGuqTjKWQkSZfFSnqajKryUOx7W_ZA92i0SUNPKRsw8bOjmA8kzkgRT__CHcNgY93FzuS5KBD3vjviK57vSEYAxUZJgQnaPQySgrsFW_WOY9gk7zoXPXcgrJBUfSIEgdbjwfu8nYus-nNnswnmCMKsUZjexNX2tDpedLKb2a7Us_UMImBJG2Uz4LGD2GUjJM9AkQwUQkqW-kENaNR3XJXcq1G3KbcHlcVcCdNXgsCzpyVMUYxN6ctXSpcSpwM3cZ5Vwcq4zdn70Yp9kCiMnVSn6bx-lnWoFwhQv991Wna_pdwfbbfTN-eWw_n8FODm7D6xxK6XiqL2y2k4rLbEx_A77a8Ow |
| 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=Algebraic+multigrid+methods+based+on+element+preconditioning&rft.jtitle=International+journal+of+computer+mathematics&rft.au=Haase%2C+G&rft.au=Langer%2C+U.&rft.au=Reitzinger%2C+S.&rft.au=Sch%C3%B6berl%2C+J.&rft.date=2001-01-01&rft.pub=Gordon+and+Breach+Science+Publishers&rft.issn=0020-7160&rft.eissn=1029-0265&rft.volume=78&rft.issue=4&rft.spage=575&rft.epage=598&rft_id=info:doi/10.1080%2F00207160108805133&rft.externalDocID=8805133 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0020-7160&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0020-7160&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0020-7160&client=summon |