{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:44:07Z","timestamp":1776840247592,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,3,13]],"date-time":"2007-03-13T00:00:00Z","timestamp":1173744000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart\u2013Thomas finite element method with a reduction factor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho &gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    uniformly for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01829-1","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1033-1042","source":"Crossref","is-referenced-by-count":62,"title":["Error reduction and convergence for an adaptive mixed finite element method"],"prefix":"10.1090","volume":"75","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[]},{"given":"R.","family":"Hoppe","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,13]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s002110050222","article-title":"Error estimators for a mixed method","volume":"74","author":"Alonso, A.","year":"1996","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502760.001.0001","volume-title":"The finite element method and its reliability","author":"Babu\u0161ka, Ivo","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0198502761"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"1207","DOI":"10.1137\/S0036142901392134","article-title":"An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition","volume":"40","author":"B\u00e4nsch, Eberhard","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","unstructured":"Bahriawati, C. and Carstensen, C. Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Newton Institute Preprint NI03069-CPD available at http:\/\/www.newton .cam.ac.uk\/preprints\/NI03069.pdf"},{"key":"6","series-title":"Lectures in Mathematics ETH Z\\\"{u}rich","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7605-6","volume-title":"Adaptive finite element methods for differential equations","author":"Bangerth, Wolfgang","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3764370092"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s00211-003-0492-7","article-title":"Adaptive finite element methods with convergence rates","volume":"97","author":"Binev, Peter","year":"2004","journal-title":"Numer. 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Low order conforming, nonconforming, and mixed FEM","volume":"71","author":"Carstensen, Carsten","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"1106","DOI":"10.1137\/0733054","article-title":"A convergent adaptive algorithm for Poisson\u2019s equation","volume":"33","author":"D\u00f6rfler, Willy","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"12","isbn-type":"print","volume-title":"Computational differential equations","author":"Eriksson, K.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0521563127"},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"1658","DOI":"10.1137\/S0036142994276992","article-title":"Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems","volume":"34","author":"Hoppe, Ronald H. W.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1137\/0722029","article-title":"An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method","volume":"22","author":"Marini, Luisa Donatella","year":"1985","journal-title":"SIAM J. Numer. 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Wiley-Teubner, New York, Stuttgart."},{"key":"18","unstructured":"Wohlmuth, B. and Hoppe, R.H.W. (1999). A comparison of a posteriori error estimators for mixed finite element discretizations. Math. Comp., 82, 253\u2013279."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01829-1\/S0025-5718-06-01829-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01829-1\/S0025-5718-06-01829-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:35:29Z","timestamp":1776782129000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01829-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3,13]]},"references-count":18,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01829-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01829-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,3,13]]}}}