{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T08:21:17Z","timestamp":1778746877254,"version":"3.51.4"},"reference-count":13,"publisher":"Wiley","issue":"2-3","license":[{"start":{"date-parts":[[2010,1,21]],"date-time":"2010-01-21T00:00:00Z","timestamp":1264032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2010,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Adaptive local refinement (ALR) can substantially improve the performance of simulations that involve numerical solution of partial differential equations. In fact, local refinement capabilities are one of the attributes of first\u2010order system least squares (FOSLS) in that it provides an inexpensive but effective <jats:italic>a posteriori<\/jats:italic> local error bound that accurately identifies regions that require further refinement. Previous theory on FOSLS established the effectiveness of its local error estimator, but only under the assumption that the local region is not too \u2018thin\u2019. This paper extends this theory to the case of a rectangular domain by showing that the estimator's effectiveness holds even for certain \u2018thin\u2019 local regions. Further, we prove that when the approximation satisfies a <jats:italic>local saturation<\/jats:italic> property, convergence of a FOSLS ALR scheme is guaranteed. Copyright \u00a9 2010 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.696","type":"journal-article","created":{"date-parts":[[2010,1,21]],"date-time":"2010-01-21T09:13:11Z","timestamp":1264065191000},"page":"387-413","source":"Crossref","is-referenced-by-count":7,"title":["Further results on error estimators for local refinement with first\u2010order system least squares (FOSLS)"],"prefix":"10.1002","volume":"17","author":[{"given":"Thomas","family":"Manteuffel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steven","family":"McCormick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joshua","family":"Nolting","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Ruge","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Geoff","family":"Sanders","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,1,21]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"35","article-title":"Local error estimates and adaptive refinement for first\u2010order system least squares (FOSLS)","volume":"6","author":"Berndt M","year":"1998","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"e_1_2_1_3_2","unstructured":"NoltingJ.Efficiency\u2010based local adaptive refinement for FOSLS finite elements. Ph.D. Thesis Applied Mathematics Department University of Colorado 2008."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/nla.567"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299527299X"},{"key":"e_1_2_1_6_2","volume-title":"p\u2010 and hp\u2010Finite Element Methods","author":"Schwab CH","year":"1998"},{"key":"e_1_2_1_7_2","volume-title":"Applied Analysis","author":"Hunter J","year":"2005"},{"key":"e_1_2_1_8_2","volume-title":"Partial Differential Equations of Mathematical Physics and Integral Equations","author":"Guenther RB","year":"1988"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/b105056"},{"key":"e_1_2_1_11_2","unstructured":"NochettoRH.Notes on design and convergence of AFEM. Lecture Notes Department of Mathematics Institute for Physical Science and Technology University of Maryland 2004."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903425082"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999360044"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1137\/0733054"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.696","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.696","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.696","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T07:09:26Z","timestamp":1693465766000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.696"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,21]]},"references-count":13,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["10.1002\/nla.696"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/nla.696","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1,21]]}}}