{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:16:30Z","timestamp":1764569790843,"version":"3.40.5"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["11170050"],"award-info":[{"award-number":["11170050"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general\nsecond-order elliptic problems.\nUnder some assumptions on the regularity of solutions of the model problem and its adjoint, superconvergence for the\nscalar field variable is achieved by either increasing the polynomial degree in the corresponding approximation space by\none or by a local postprocessing.\nWe provide a uniform analysis that allows the treatment of different test norms.\nParticularly, we show that in the presence of convection only the quasi-optimal test norm leads to higher convergence\nrates, whereas other norms considered do not.\nMoreover, we also prove that our DPG method delivers the best <jats:inline-formula id=\"j_cmam-2018-0250_ineq_9999_w2aab3b7e4835b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0250_eq_0351.png\"\/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> approximation of the scalar field variable\nup to higher-order terms, which is the first theoretical explanation of an observation made previously by different authors.\nNumerical studies that support our theoretical findings are presented.<\/jats:p>","DOI":"10.1515\/cmam-2018-0250","type":"journal-article","created":{"date-parts":[[2019,4,7]],"date-time":"2019-04-07T02:48:35Z","timestamp":1554605315000},"page":"483-502","source":"Crossref","is-referenced-by-count":13,"title":["Superconvergent DPG Methods for Second-Order Elliptic Problems"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5034-6593","authenticated-orcid":false,"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Santiago , Chile"}]}],"member":"374","published-online":{"date-parts":[[2019,4,6]]},"reference":[{"key":"2023033110340742719_j_cmam-2018-0250_ref_001_w2aab3b7e4835b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"T.  Bouma, J.  Gopalakrishnan and A.  Harb,\nConvergence rates of the DPG method with reduced test space degree,\nComput. Math. Appl. 68 (2014), no. 11, 1550\u20131561.","DOI":"10.1016\/j.camwa.2014.08.004"},{"key":"2023033110340742719_j_cmam-2018-0250_ref_002_w2aab3b7e4835b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, L.  Demkowicz and J.  Gopalakrishnan,\nA posteriori error control for DPG methods,\nSIAM J. Numer. Anal. 52 (2014), no. 3, 1335\u20131353.","DOI":"10.1137\/130924913"},{"key":"2023033110340742719_j_cmam-2018-0250_ref_003_w2aab3b7e4835b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, L.  Demkowicz and J.  Gopalakrishnan,\nBreaking spaces and forms for the DPG method and applications including Maxwell equations,\nComput. Math. Appl. 72 (2016), no. 3, 494\u2013522.","DOI":"10.1016\/j.camwa.2016.05.004"},{"key":"2023033110340742719_j_cmam-2018-0250_ref_004_w2aab3b7e4835b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"B.  Cockburn, B.  Dong and J.  Guzm\u00e1n,\nA superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems,\nMath. Comp. 77 (2008), no. 264, 1887\u20131916.","DOI":"10.1090\/S0025-5718-08-02123-6"},{"key":"2023033110340742719_j_cmam-2018-0250_ref_005_w2aab3b7e4835b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and J.  Gopalakrishnan,\nA class of discontinuous Petrov\u2013Galerkin methods. Part I: The transport equation,\nComput. Methods Appl. Mech. Engrg. 199 (2010), no. 23\u201324, 1558\u20131572.","DOI":"10.1016\/j.cma.2010.01.003"},{"key":"2023033110340742719_j_cmam-2018-0250_ref_006_w2aab3b7e4835b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and J.  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