{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T04:33:40Z","timestamp":1772858020248,"version":"3.50.1"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1210391"],"award-info":[{"award-number":["1210391"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>H<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>\u2212<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0215_ineq_0001.png\"\/>\n                        <jats:tex-math>H^{-1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> loads.\nWe prove that any bounded <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>H<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>\u2212<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0215_ineq_0001.png\"\/>\n                        <jats:tex-math>H^{-1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms.\nExamples for the construction of such projectors are given.\nOne is based on the adjoint of a weighted Cl\u00e9ment quasi-interpolator.\nWe prove that this Cl\u00e9ment operator has second-order approximation properties.\nFor the modified mixed method, we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions\u2014a result not valid for the lowest-order mixed FEM without regularization.\nNumerical examples conclude this work.<\/jats:p>","DOI":"10.1515\/cmam-2022-0215","type":"journal-article","created":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T18:06:13Z","timestamp":1683569173000},"page":"363-378","source":"Crossref","is-referenced-by-count":6,"title":["On a Mixed FEM and a FOSLS with \ud835\udc3b<sup>\u22121<\/sup> Loads"],"prefix":"10.1515","volume":"24","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,5,9]]},"reference":[{"key":"2025062712372351052_j_cmam-2022-0215_ref_001","doi-asserted-by":"crossref","unstructured":"F. Bertrand and D. Boffi,\nFirst order least-squares formulations for eigenvalue problems,\nIMA J. Numer. Anal. 42 (2022), no. 2, 1339\u20131363.","DOI":"10.1093\/imanum\/drab005"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_002","doi-asserted-by":"crossref","unstructured":"P. B. Bochev and M. D. Gunzburger,\nLeast-Squares Finite Element Methods,\nAppl. Math. Sci. 166,\nSpringer, New York, 2009.","DOI":"10.1007\/b13382"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_003","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_004","unstructured":"P. Bringmann,\nComputational competition of three adaptive least-squares finite element schemes,\npreprint (2022), https:\/\/arxiv.org\/abs\/2209.06028."},{"key":"2025062712372351052_j_cmam-2022-0215_ref_005","doi-asserted-by":"crossref","unstructured":"Z. Cai and J. Ku,\nOptimal error estimate for the div least-squares method with data \n                  \n                     \n                        \n                           f\n                           \u2208\n                           \n                              L\n                              2\n                           \n                        \n                     \n                     \n                     f\\in L^{2}\n                  \n                and application to nonlinear problems,\nSIAM J. Numer. Anal. 47 (2010), no. 6, 4098\u20134111.","DOI":"10.1137\/080738350"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_006","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nQuasi-interpolation and a posteriori error analysis in finite element methods,\nM2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187\u20131202.","DOI":"10.1051\/m2an:1999140"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_007","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nCl\u00e9ment interpolation and its role in adaptive finite element error control,\nPartial Differential Equations and Functional Analysis,\nOper. Theory Adv. Appl. 168,\nBirkh\u00e4user, Basel (2006), 27\u201343.","DOI":"10.1007\/3-7643-7601-5_2"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_008","doi-asserted-by":"crossref","unstructured":"P. Cl\u00e9ment,\nApproximation by finite element functions using local regularization,\nRev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. 9 (1975), no. R-2, 77\u201384.","DOI":"10.1051\/m2an\/197509R200771"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_009","doi-asserted-by":"crossref","unstructured":"M. Dauge,\nElliptic Boundary Value Problems on Corner Domains,\nLecture Notes in Math. 1341,\nSpringer, Berlin, 1988.","DOI":"10.1007\/BFb0086682"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_010","doi-asserted-by":"crossref","unstructured":"L. Diening, J. Storn and T. Tscherpel,\nInterpolation operator on negative Sobolev spaces,\nMath. Comp. 92 (2023), 1511\u20131541.","DOI":"10.1090\/mcom\/3824"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_011","doi-asserted-by":"crossref","unstructured":"A. Ern, T. Gudi, I. Smears and M. Vohral\u00edk,\nEquivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n                approximation estimates in \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              div\n                              )\n                           \n                        \n                     \n                     \n                     \\mathbf{H}(\\mathrm{div})\n                  \n               ,\nIMA J. Numer. Anal. 42 (2022), no. 2, 1023\u20131049.","DOI":"10.1093\/imanum\/draa103"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_012","doi-asserted-by":"crossref","unstructured":"A. Ern and P. Zanotti,\nA quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with \n                  \n                     \n                        \n                           H\n                           \n                              \u2212\n                              1\n                           \n                        \n                     \n                     \n                     H^{-1}\n                  \n                loads,\nIMA J. Numer. Anal. 40 (2020), no. 4, 2163\u20132188.","DOI":"10.1093\/imanum\/drz057"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_013","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer,\nSuperconvergent DPG methods for second-order elliptic problems,\nComput. Methods Appl. Math. 19 (2019), no. 3, 483\u2013502.","DOI":"10.1515\/cmam-2018-0250"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_014","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer,\nMultilevel decompositions and norms for negative order Sobolev spaces,\nMath. Comp. 91 (2021), no. 333, 183\u2013218.","DOI":"10.1090\/mcom\/3674"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_015","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer, N. Heuer and M. Karkulik,\nMINRES for second-order PDEs with singular data,\nSIAM J. Numer. Anal. 60 (2022), no. 3, 1111\u20131135.","DOI":"10.1137\/21M1457023"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_016","doi-asserted-by":"crossref","unstructured":"G. N. Gatica,\nA Simple Introduction to the Mixed Finite Element Method,\nSpringer Briefs Math.,\nSpringer, Cham, 2014.","DOI":"10.1007\/978-3-319-03695-3"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_017","unstructured":"P. Grisvard,\nElliptic Problems in Nonsmooth Domains,\nMonogr. Stud. Math. 24,\nPitman, Boston, 1985."},{"key":"2025062712372351052_j_cmam-2022-0215_ref_018","doi-asserted-by":"crossref","unstructured":"J. Ku,\nSharp \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L_{2}\n                  \n               -norm error estimates for first-order div least-squares methods,\nSIAM J. Numer. Anal. 49 (2011), no. 2, 755\u2013769.","DOI":"10.1137\/100792470"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_019","doi-asserted-by":"crossref","unstructured":"F. Millar, I. Muga, S. Rojas and K. G. Van der Zee,\nProjection in negative norms and the regularization of rough linear functionals,\nNumer. Math. 150 (2022), no. 4, 1087\u20131121.","DOI":"10.1007\/s00211-022-01278-z"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_020","doi-asserted-by":"crossref","unstructured":"H. Monsuur, R. Stevenson and J. Storn,\nMinimal residual methods in negative or fractional sobolev norms,\npreprint (2023), https:\/\/arxiv.org\/abs\/2301.10484.","DOI":"10.1090\/mcom\/3904"},{"key":"2025062712372351052_j_cmam-2022-0215_ref_021","unstructured":"I. Muga, S. Rojas and P. Vega,\nAn adaptive superconvergent finite element method based on local residual minimization,\npreprint (2022), https:\/\/arxiv.org\/abs\/2210.00390."},{"key":"2025062712372351052_j_cmam-2022-0215_ref_022","doi-asserted-by":"crossref","unstructured":"R. Stenberg,\nPostprocessing schemes for some mixed finite elements,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 25 (1991), no. 1, 151\u2013167.","DOI":"10.1051\/m2an\/1991250101511"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0215\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0215\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T12:38:20Z","timestamp":1751027900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0215\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,9]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,1,2]]},"published-print":{"date-parts":[[2024,4,1]]}},"alternative-id":["10.1515\/cmam-2022-0215"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0215","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,9]]}}}