{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:41:44Z","timestamp":1776724904079,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"228","license":[{"start":{"date-parts":[[2000,5,25]],"date-time":"2000-05-25T00:00:00Z","timestamp":959212800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in Campanato spaces. The proofs rely on discrete analogues of the Campanato inequalities for the solution of the system, which locally measure the decay of the energy. As an application of our results we derive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W Superscript 1 comma p\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">W^{1,p}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013estimates and give a new proof of the well-known\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W Superscript 1 comma normal infinity\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">W^{1,\\infty }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013results of Rannacher and Scott.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01175-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1397-1427","source":"Crossref","is-referenced-by-count":6,"title":["Optimal convergence for the finite element method in Campanato spaces"],"prefix":"10.1090","volume":"68","author":[{"given":"Georg","family":"Dolzmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,25]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/BF02576116","article-title":"Regularity results for discrete solutions of second order elliptic problems in the finite element method","volume":"23","author":"Aguilera, N. E.","year":"1986","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01389427","article-title":"Asymptotic error expansion and Richardson extrapolation for linear finite elements","volume":"49","author":"Blum, H.","year":"1986","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"key":"4","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF02414377","article-title":"Equazioni ellittiche del \ud835\udc3c\ud835\udc3cdeg ordine espazi \ud835\udd0f^{(2,\ud835\udf06)}","volume":"69","author":"Campanato, Sergio","year":"1965","journal-title":"Ann. Mat. Pura Appl. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4622","issn-type":"print"},{"key":"5","series-title":"Quaderni. [Publications]","volume-title":"Sistemi ellittici in forma divergenza. Regolarit\\`a all'interno","author":"Campanato, S.","year":"1980"},{"key":"6","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":"7","series-title":"Bonner Mathematische Schriften [Bonn Mathematical Publications]","volume-title":"Campanato-Ungleichungen f\\\"{u}r Differenzenverfahren und finite Elemente","volume":"251","author":"Dolzmann, Georg","year":"1993"},{"key":"8","first-page":"95","article-title":"Optimale asymptotische Konvergenz linearer finiter Elemente bei elliptischen Problemen mit Hilfe von Campanato-Absch\u00e4tzungen","author":"Dolzmann, G.","year":"1991"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/BF02567822","article-title":"Estimates for Green\u2019s matrices of elliptic systems by \ud835\udc3f^{\ud835\udc5d} theory","volume":"88","author":"Dolzmann, G.","year":"1995","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"issue":"179","key":"10","doi-asserted-by":"publisher","first-page":"17","DOI":"10.2307\/2008247","article-title":"Quasi-optimal estimates for finite element approximations using Orlicz norms","volume":"49","author":"Dur\u00e1n, Ricardo G.","year":"1987","journal-title":"Math. 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Anwendungen","ISSN":"https:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"key":"13","series-title":"Annals of Mathematics Studies","isbn-type":"print","volume-title":"Multiple integrals in the calculus of variations and nonlinear elliptic systems","volume":"105","author":"Giaquinta, Mariano","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0691083304"},{"key":"14","unstructured":"[Gi] E. Giusti, Metodi diretti nel calcolo delle variazioni, Unione Matematica Italiana, Bologna, 1994."},{"key":"15","first-page":"535","article-title":"A new definition of uniform spaces","volume":"135","author":"Sandberg, V. Ju.","journal-title":"Dokl. Akad. 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Nitsche, Schauder Estimates for Finite Element Approximations on Second Order Elliptic Boundary Value Problems, Proceedings of a Special Year in Numerical Analysis, University of Maryland, Numerical Analysis Lecture Notes 20, 1981."},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF01301633","article-title":"Zur \ud835\udc3f^{\u221e}-Konvergenz linearer finiter Elemente beim Dirichlet-Problem","volume":"149","author":"Rannacher, Rolf","year":"1976","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"158","key":"20","doi-asserted-by":"publisher","first-page":"437","DOI":"10.2307\/2007280","article-title":"Some optimal error estimates for piecewise linear finite element approximations","volume":"38","author":"Rannacher, Rolf","year":"1982","journal-title":"Math. 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