{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:40:54Z","timestamp":1680266454771},"reference-count":18,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100002848","name":"Comisi\u00f3n Nacional de Investigaci\u00f3n Cient\u00edfica y Tecnol\u00f3gica","doi-asserted-by":"publisher","award":["1150056","1170672","ACT1118","PFB-03"],"award-info":[{"award-number":["1150056","1170672","ACT1118","PFB-03"]}],"id":[{"id":"10.13039\/501100002848","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We propose and analyze a discretization scheme that combines the discontinuous Petrov\u2013Galerkin\nand finite element methods. The underlying model problem is of general diffusion-advection-reaction\ntype on bounded domains, with decomposition into two sub-domains. We propose\na heterogeneous variational formulation that is of the ultra-weak (Petrov\u2013Galerkin)\nform with broken test space in one part, and of Bubnov\u2013Galerkin form in the other.\nA standard discretization with conforming approximation spaces and appropriate test spaces\n(optimal test functions for the ultra-weak part and standard test functions for the Bubnov\u2013Galerkin\npart) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal\nconvergence. Numerical results confirm expected convergence orders.<\/jats:p>","DOI":"10.1515\/cmam-2017-0041","type":"journal-article","created":{"date-parts":[[2017,9,30]],"date-time":"2017-09-30T10:01:39Z","timestamp":1506765699000},"page":"639-652","source":"Crossref","is-referenced-by-count":0,"title":["Combining the DPG Method with Finite Elements"],"prefix":"10.1515","volume":"18","author":[{"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Avenida Vicu\u00f1a Mackenna 4860 , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Norbert","family":"Heuer","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Avenida Vicu\u00f1a Mackenna 4860 , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Karkulik","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica , Universidad T\u00e9cnica Federico Santa Mar\u00eda , Avenida Espa\u00f1a 1680 , Valpara\u00edso , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodolfo","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda Matem\u00e1tica and Centro de Investigaci\u00f3n en Ingenier\u00eda Matem\u00e1tica , Universidad de Concepci\u00f3n , Casilla 160-C , Concepci\u00f3n , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,30]]},"reference":[{"key":"2023033110390265100_j_cmam-2017-0041_ref_001_w2aab3b7d531b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"J. W.  Barrett and K. W.  Morton,\nApproximate symmetrization and Petrov\u2013Galerkin methods for diffusion-convection problems,\nComput. Methods Appl. Mech. Engrg. 45 (1984), no. 1\u20133, 97\u2013122.","DOI":"10.1016\/0045-7825(84)90152-X"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_002_w2aab3b7d531b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"D.  Broersen and R.  Stevenson,\nA robust Petrov\u2013Galerkin discretisation of convection-diffusion equations,\nComput. Math. Appl. 68 (2014), no. 11, 1605\u20131618.","DOI":"10.1016\/j.camwa.2014.06.019"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_003_w2aab3b7d531b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"D.  Broersen and R. P.  Stevenson,\nA Petrov\u2013Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form,\nIMA J. Numer. Anal. 35 (2015), no. 1, 39\u201373.","DOI":"10.1093\/imanum\/dru003"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_004_w2aab3b7d531b1b6b1ab2ab4Aa","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":"2023033110390265100_j_cmam-2017-0041_ref_005_w2aab3b7d531b1b6b1ab2ab5Aa","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":"2023033110390265100_j_cmam-2017-0041_ref_006_w2aab3b7d531b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"P.  Causin and R.  Sacco,\nA discontinuous Petrov\u2013Galerkin method with Lagrangian multipliers for second order elliptic problems,\nSIAM J. Numer. Anal. 43 (2005), no. 1, 280\u2013302.","DOI":"10.1137\/S0036142903427871"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_007_w2aab3b7d531b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"J.  Chan, N.  Heuer, T.  Bui-Thanh and L.  Demkowicz,\nA robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms,\nComput. Math. Appl. 67 (2014), no. 4, 771\u2013795.","DOI":"10.1016\/j.camwa.2013.06.010"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_008_w2aab3b7d531b1b6b1ab2ab8Aa","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":"2023033110390265100_j_cmam-2017-0041_ref_009_w2aab3b7d531b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and J.  Gopalakrishnan,\nA class of discontinuous Petrov\u2013Galerkin methods. II. Optimal test functions,\nNumer. Methods Partial Differential Equations 27 (2011), no. 1, 70\u2013105.","DOI":"10.1002\/num.20640"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_010_w2aab3b7d531b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz, J.  Gopalakrishnan and A. H.  Niemi,\nA class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity,\nAppl. Numer. Math. 62 (2012), no. 4, 396\u2013427.","DOI":"10.1016\/j.apnum.2011.09.002"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_011_w2aab3b7d531b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and N.  Heuer,\nRobust DPG method for convection-dominated diffusion problems,\nSIAM J. Numer. Anal. 51 (2013), no. 5, 2514\u20132537.","DOI":"10.1137\/120862065"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_012_w2aab3b7d531b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"F.  Fuentes, B.  Keith, L.  Demkowicz and P.  Le Tallec,\nCoupled variational formulations of linear elasticity and the DPG methodology,\nJ. Comput. Phys. 348 (2017), 715\u2013731.","DOI":"10.1016\/j.jcp.2017.07.051"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_013_w2aab3b7d531b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer and N.  Heuer,\nRobust coupling of DPG and BEM for a singularly perturbed transmission problem,\nComput. Math. Appl. (2016), 10.1016\/j.camwa.2016.09.016.","DOI":"10.1090\/mcom\/3170"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_014_w2aab3b7d531b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer, N.  Heuer and M.  Karkulik,\nOn the coupling of DPG and BEM,\nMath. Comp. 86 (2017), no. 307, 2261\u20132284.","DOI":"10.1090\/mcom\/3170"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_015_w2aab3b7d531b1b6b1ab2ac15Aa","unstructured":"T.  F\u00fchrer, N.  Heuer and E. P.  Stephan,\nOn the DPG method for Signorini problems,\npreprint (2016), https:\/\/arxiv.org\/abs\/1609.00765;\nto appear in IMA J. Numer. Anal., DOI 10.1093\/imanum\/drx048."},{"key":"2023033110390265100_j_cmam-2017-0041_ref_016_w2aab3b7d531b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"J.  Gopalakrishnan and W.  Qiu,\nAn analysis of the practical DPG method,\nMath. Comp. 83 (2014), no. 286, 537\u2013552.","DOI":"10.1090\/S0025-5718-2013-02721-4"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_017_w2aab3b7d531b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"N.  Heuer and M.  Karkulik,\nA robust DPG method for singularly perturbed reaction-diffusion problems,\nSIAM J. Numer. Anal. 55 (2017), no. 3, 1218\u20131242.","DOI":"10.1137\/15M1041304"},{"key":"2023033110390265100_j_cmam-2017-0041_ref_018_w2aab3b7d531b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"N. V.  Roberts,\nCamellia: A software framework for discontinuous Petrov\u2013Galerkin methods,\nComput. Math. 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