{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:04:56Z","timestamp":1776769496928,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"235","license":[{"start":{"date-parts":[[2001,3,24]],"date-time":"2001-03-24T00:00:00Z","timestamp":985392000000},"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                    In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in\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                    and in the broken\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm for the internal variable\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n                        <mml:semantics>\n                          <mml:mi>u<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and in suitable weighted\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                    norms for the other variables\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03bb\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c8\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01250-3","type":"journal-article","created":{"date-parts":[[2002,11,6]],"date-time":"2002-11-06T14:02:22Z","timestamp":1036591342000},"page":"911-934","source":"Crossref","is-referenced-by-count":31,"title":["Error estimates for the three-field formulation with bubble stabilization"],"prefix":"10.1090","volume":"70","author":[{"given":"Franco","family":"Brezzi","sequence":"first","affiliation":[]},{"given":"Donatella","family":"Marini","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,3,24]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/0045-7825(93)90119-I","article-title":"Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.)","volume":"105","author":"Baiocchi, Claudio","year":"1993","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"2","isbn-type":"print","first-page":"345","article-title":"Stabilization of Galerkin methods and applications to domain decomposition","author":"Baiocchi, C.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3540563202"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1137\/0723075","article-title":"Iterative methods for the solution of elliptic problems on regions partitioned into substructures","volume":"23","author":"Bj\u00f8rstad, Petter E.","year":"1986","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"4","unstructured":"P.E. Bj\u00f8rstad, M.S. Espedal and D.E. Keyes (eds.), Domain Decomposition Methods in Science and Engineering, Domain Decomposition Press, Bergen, 1998."},{"issue":"187","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008346","article-title":"The construction of preconditioners for elliptic problems by substructuring. IV","volume":"53","author":"Bramble, James H.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","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":"7","unstructured":"F. Brezzi, L.P. Franca, L.D. Marini, and A. Russo, Stabilization techniques for domain decomposition methods with non-matching grids, in [4], pp. 1\u201311."},{"key":"8","isbn-type":"print","first-page":"89","article-title":"Macro hybrid elements and domain decomposition methods","author":"Brezzi, F.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/2854283325"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1090\/conm\/157\/01402","article-title":"A three-field domain decomposition method","author":"Brezzi, F.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821851586"},{"key":"10","unstructured":"A. Buffa, Error estimate for a stabilized domain decomposition method with nonmatching grid. Numer. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"key":"17","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/BF02995904","article-title":"\u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet-Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind","volume":"36","author":"Nitsche, J.","year":"1971","journal-title":"Abh. Math. Sem. Univ. Hamburg","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5858","issn-type":"print"},{"issue":"6","key":"18","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.1137\/0911069","article-title":"A domain decomposition algorithm using a hierarchical basis","volume":"11","author":"Smith, Barry F.","year":"1990","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"issue":"6","key":"19","doi-asserted-by":"publisher","first-page":"1486","DOI":"10.1137\/0912082","article-title":"A domain decomposition preconditioner based on a change to a multilevel nodal basis","volume":"12","author":"Tong, Charles H.","year":"1991","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"20","unstructured":"J. Xu, Theory of multilevel methods, PhD thesis, Cornell University, 1989."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01250-3\/S0025-5718-00-01250-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01250-3\/S0025-5718-00-01250-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:40:44Z","timestamp":1776724844000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01250-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3,24]]},"references-count":20,"journal-issue":{"issue":"235","published-print":{"date-parts":[[2001,7]]}},"alternative-id":["S0025-5718-00-01250-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01250-3","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":[[2000,3,24]]}}}