Goal-oriented error control of the iterative solution of finite element equations
This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel...
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| Published in | Journal of numerical mathematics Vol. 17; no. 2; pp. 143 - 172 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Walter de Gruyter GmbH & Co. KG
01.07.2009
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1570-2820 1569-3953 |
| DOI | 10.1515/JNUM.2009.009 |
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| Abstract | This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems. |
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| AbstractList | This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems. |
| Author | Meidner, D. Rannacher, R. Vihharev, J. |
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| CitedBy_id | crossref_primary_10_1515_cmam_2020_0027 crossref_primary_10_1007_s10915_014_9921_2 crossref_primary_10_1002_gamm_201310006 crossref_primary_10_1515_cmam_2021_0030 crossref_primary_10_1016_j_cma_2013_12_001 crossref_primary_10_1007_s11075_021_01118_5 crossref_primary_10_1016_j_cam_2019_112367 crossref_primary_10_1137_19M1275929 crossref_primary_10_1515_jnum_2010_015 crossref_primary_10_3934_dcdss_2012_5_1147 crossref_primary_10_1137_120896918 crossref_primary_10_1007_s10208_010_9082_0 crossref_primary_10_1007_s00211_017_0925_3 crossref_primary_10_1016_j_camwa_2020_01_005 crossref_primary_10_1002_pamm_202400157 crossref_primary_10_1016_j_apnum_2019_05_016 crossref_primary_10_1016_j_cma_2015_01_009 crossref_primary_10_1016_j_cam_2009_07_026 crossref_primary_10_1515_cmam_2020_0036 crossref_primary_10_1007_s10915_020_01188_y crossref_primary_10_1137_08073706X crossref_primary_10_1016_j_jcp_2017_01_037 crossref_primary_10_1016_j_ocemod_2020_101684 crossref_primary_10_1515_cmam_2018_0010 crossref_primary_10_5802_crmeca_160 crossref_primary_10_1007_s10596_019_09909_5 crossref_primary_10_1007_s00466_023_02427_3 crossref_primary_10_1007_s10915_024_02485_6 crossref_primary_10_1007_s10543_020_00822_3 crossref_primary_10_1016_j_cma_2020_113243 crossref_primary_10_1137_18M1227275 crossref_primary_10_1016_j_jcp_2017_09_033 |
| Cites_doi | 10.1002/1099-1506(200009)7:6<363::AID-NLA202>3.0.CO;2-V 10.1137/S003614299732334X 10.1007/BF02238483 10.2307/2007724 10.2307/2153097 10.2307/2006422 10.1007/s00211-003-0482-9 10.1137/S1540345902410482 10.2307/2938661 |
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| SubjectTerms | a posteriori error estimation finite elements goal-oriented adaptivity iterative methods multigrid |
| Title | Goal-oriented error control of the iterative solution of finite element equations |
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