{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T22:36:29Z","timestamp":1774478189509,"version":"3.50.1"},"reference-count":31,"publisher":"Wiley","license":[{"start":{"date-parts":[[2009,7,12]],"date-time":"2009-07-12T00:00:00Z","timestamp":1247356800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["SFB F013\/F1306"],"award-info":[{"award-number":["SFB F013\/F1306"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Numerical Analysis"],"published-print":{"date-parts":[[2009,7,12]]},"abstract":"<jats:p>We consider a Poisson boundary value problem and its functional a posteriori error estimate derived by S. Repin in 1999. The estimate majorizes the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msup><mml:mi>H<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math> seminorm of the error of the discrete solution computed by FEM method and contains a free ux variable from the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>H<\/mml:mi><\/mml:math>(div) space. In order to keep the estimate sharp, a procedure for the minimization of the majorant term with respect to the ux variable is introduced, computing the free ux variable from a global linear system of equations. Since the linear system is symmetric and positive definite, few iterations of a conjugate gradient method with a geometrical multigrid preconditioner are applied. Numerical techniques are demonstated on one benchmark example with a smooth solution on a unit square domain including the computation of the approximate value of the constant in Friedrichs' inequality.<\/jats:p>","DOI":"10.1155\/2009\/164519","type":"journal-article","created":{"date-parts":[[2009,7,12]],"date-time":"2009-07-12T13:16:09Z","timestamp":1247404569000},"page":"1-15","source":"Crossref","is-referenced-by-count":10,"title":["Minimization of Functional Majorant in a Posteriori Error Analysis Based on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>H<\/mml:mi><mml:mtext>(div)<\/mml:mtext><\/mml:math>  Multigrid-Preconditioned CG Method"],"prefix":"10.1155","volume":"2009","author":[{"given":"Jan","family":"Valdman","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Bergen, Johannes Bruns Gate 12, 5008 Bergen, Norway"}]}],"member":"311","reference":[{"key":"12","series-title":"Studies in Mathematics and Its 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