{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:38:09Z","timestamp":1776847089440,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"263","license":[{"start":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T00:00:00Z","timestamp":1236124800000},"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                    Superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes. For a large class of practically useful grids, the finite element solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">u_h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is proven to be superclose to the interpolant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript upper I\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mi>I<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">u_I<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and as a result a postprocessing gradient recovery scheme for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">u_h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be devised. The analysis is based on a number of carefully derived identities. In addition to its own theoretical interests, the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02051-6","type":"journal-article","created":{"date-parts":[[2008,4,22]],"date-time":"2008-04-22T17:24:54Z","timestamp":1208885094000},"page":"1253-1268","source":"Crossref","is-referenced-by-count":40,"title":["Superconvergence of quadratic finite elements on mildly structured grids"],"prefix":"10.1090","volume":"77","author":[{"given":"Yunqing","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinchao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,3,4]]},"reference":[{"issue":"9","key":"1","first-page":"1179","article-title":"Error estimate of type superconvergence of the gradient for quadratic triangular elements","volume":"37","author":"Andreev, A. B.","year":"1984","journal-title":"C. R. Acad. Bulgare Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0366-8681","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1002\/num.1690040103","article-title":"Superconvergence of the gradient for quadratic triangular finite elements","volume":"4","author":"Andreev, A. B.","year":"1988","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"I. Babuska and W. Rheinboldt. A posteriori error estimates for the finite element method. Internat. J. Numer. 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Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0168-9274(99)00034-3","article-title":"Superconvergence for triangular order \ud835\udc58=1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods","volume":"34","author":"Brandts, Jan H.","year":"2000","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"8","unstructured":"C. M. Chen and Y. Huang. High accuracy theory of finite element methods. Hunan, Science Press, Hunan, China (in Chinese), 1995."},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1002\/num.1690070106","article-title":"Superconvergence of recovered gradients of piecewise quadratic finite element approximations. I. \ud835\udc3f\u2082-error estimates","volume":"7","author":"Goodsell, G.","year":"1991","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"228","key":"10","doi-asserted-by":"publisher","first-page":"1325","DOI":"10.1090\/S0025-5718-99-01166-7","article-title":"Canonical construction of finite elements","volume":"68","author":"Hiptmair, R.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"11","first-page":"75","article-title":"Superconvergence for higher-order triangular finite elements","volume":"12","author":"Li, Bo","year":"1990","journal-title":"Chinese J. Numer. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0899-4358","issn-type":"print"},{"key":"12","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0096835","volume-title":"Superconvergence in Galerkin finite element methods","volume":"1605","author":"Wahlbin, Lars B.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3540600116"},{"key":"13","unstructured":"Jinchao Xu and Z. M. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp., pages 781\u2013801, 2003."},{"issue":"7","key":"14","doi-asserted-by":"publisher","first-page":"1321","DOI":"10.1002\/nme.1620300707","article-title":"Superconvergence recovery technique and a posteriori error estimators","volume":"30","author":"Zhu, J. Z.","year":"1990","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"15","unstructured":"Q. Zhu. The derivative good points for the finite element method with 2-degree triangular element (in chinese) Natural Science Journal of Xiangtan University, 4:36\u201345, 1981."},{"key":"16","isbn-type":"print","first-page":"935","article-title":"Natural inner superconvergence for the finite element method","author":"Zhu, Qi Ding","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0677313101"},{"key":"17","unstructured":"Q. Zhu and Q. Lin. Finite element superconvergence theory. Hunan Science Press, 1989."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02051-6\/S0025-5718-08-02051-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02051-6\/S0025-5718-08-02051-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:21:04Z","timestamp":1776784864000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02051-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,4]]},"references-count":17,"journal-issue":{"issue":"263","published-print":{"date-parts":[[2008,7]]}},"alternative-id":["S0025-5718-08-02051-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02051-6","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":[[2008,3,4]]}}}