{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T08:03:40Z","timestamp":1778400220695,"version":"3.51.4"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A computable error bound for mixed finite element methods is established in the model case of the Poisson\u2013problem to control the error in the H(div,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    )\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"times upper L squared left-parenthesis normal upper Omega right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u03a9\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\times L^2(\\Omega )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013norm. The reliable and efficient a\u00a0posteriori error estimate applies, e.g., to Raviart\u2013Thomas, Brezzi-Douglas-Marini, and Brezzi-Douglas-Fortin-Marini elements.\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00837-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"465-476","source":"Crossref","is-referenced-by-count":215,"title":["A posteriori error estimate for the mixed finite element method"],"prefix":"10.1090","volume":"66","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","unstructured":"[BV] D. Braess, R. Verf\u00fcrth: A posteriori error estimators for the Raviart-Thomas element. Preprint 175\/1994 Fakult\u00e4t f\u00fcr Mathematik der Ruhr-Universit\u00e4t Bochum."},{"key":"2","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"key":"3","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"key":"4","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1051\/m2an\/197509R200771","article-title":"Approximation by finite element functions using local regularization","volume":"9","author":"Cl\u00e9ment, Ph.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[EEHJ] K. Eriksson, D. Estep, P. Hansbo, C. Johnson:  Introduction to adaptive methods for differential equations.  Acta Numerica (1995) 105\u2014158.","DOI":"10.1017\/S0962492900002531"},{"key":"6","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"key":"7","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 116","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-46175-0","volume-title":"Linear partial differential operators","author":"H\u00f6rmander, Lars","year":"1963"},{"key":"8","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 181","volume-title":"Non-homogeneous boundary value problems and applications. Vol. I","author":"Lions, J.-L.","year":"1972"},{"key":"9","series-title":"Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics]","isbn-type":"print","volume-title":"Polygonal interface problems","volume":"39","author":"Nicaise, Serge","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3631463804"},{"key":"10","unstructured":"[V1] R. Verf\u00fcrth: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner Skripten zur Numerik. B.G. Teubner Stuttgart 1996."},{"issue":"1-3","key":"11","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/0377-0427(94)90290-9","article-title":"A posteriori error estimation and adaptive mesh-refinement techniques","volume":"50","author":"Verf\u00fcrth, R.","year":"1994","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"206","key":"12","doi-asserted-by":"publisher","first-page":"445","DOI":"10.2307\/2153518","article-title":"A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations","volume":"62","author":"Verf\u00fcrth, R.","year":"1994","journal-title":"Math. 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