{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:16:43Z","timestamp":1760267803094,"version":"3.40.5"},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/100006234","name":"Sandia National Laboratories","doi-asserted-by":"publisher","award":["218322"],"award-info":[{"award-number":["218322"]}],"id":[{"id":"10.13039\/100006234","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1819101"],"award-info":[{"award-number":["1819101"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["101017984"],"award-info":[{"award-number":["101017984"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem.\nWe determine that the most common approach for DPG stability analysis \u2013 construction of a local Fortin operator \u2013 is infeasible for the convection-reaction problem.\nWe then develop a line of argument based on a direct proof of discrete stability; we find that employing a polynomial enrichment for the test space does not suffice for this purpose, motivating the introduction of a (two-element) subgrid mesh.\nThe argument combines mathematical analysis with numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2021-0149","type":"journal-article","created":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T10:32:29Z","timestamp":1655807549000},"page":"93-125","source":"Crossref","is-referenced-by-count":2,"title":["The DPG Method for the Convection-Reaction Problem, Revisited"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7839-8037","authenticated-orcid":false,"given":"Leszek Feliks","family":"Demkowicz","sequence":"first","affiliation":[{"name":"Oden Institute , The University of Texas , Austin , USA"}]},{"given":"Nathan V.","family":"Roberts","sequence":"additional","affiliation":[{"name":"Center for Computer Research , Sandia National Laboratories , Albuquerque , USA"}]},{"given":"Judit","family":"Mu\u00f1oz-Matute","sequence":"additional","affiliation":[{"name":"Oden Institute , The University of Texas , Austin , USA ; and Basque Center for Applied Mathematics, Bilbo, Spain"}]}],"member":"374","published-online":{"date-parts":[[2022,6,22]]},"reference":[{"key":"2023033112515576898_j_cmam-2021-0149_ref_001","doi-asserted-by":"crossref","unstructured":"P. Bringmann and C. Carstensen,\nAn adaptive least-squares FEM for the Stokes equations with optimal convergence rates,\nNumer. Math. 135 (2017), no. 2, 459\u2013492.","DOI":"10.1007\/s00211-016-0806-1"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_002","doi-asserted-by":"crossref","unstructured":"D. Broersen, W. Dahmen and R. P. Stevenson,\nOn the stability of DPG formulations of transport equations,\nMath. Comp. 87 (2018), no. 311, 1051\u20131082.","DOI":"10.1090\/mcom\/3242"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_003","doi-asserted-by":"crossref","unstructured":"J. Brunken, K. Smetana and K. Urban,\n(Parametrized) first order transport equations: Realization of optimally stable Petrov\u2013Galerkin methods,\nSIAM J. Sci. Comput. 41 (2019), no. 1, A592\u2013A621.","DOI":"10.1137\/18M1176269"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_004","doi-asserted-by":"crossref","unstructured":"T. Bui-Thanh, L. Demkowicz and O. Ghattas,\nA unified discontinuous Petrov\u2013Galerkin method and its analysis for Friedrichs\u2019 systems,\nSIAM J. Numer. Anal. 51 (2013), no. 4, 1933\u20131958.","DOI":"10.1137\/110854369"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_005","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":"2023033112515576898_j_cmam-2021-0149_ref_006","doi-asserted-by":"crossref","unstructured":"C. Carstensen and F. Hellwig,\nLow-order discontinuous Petrov\u2013Galerkin finite element methods for linear elasticity,\nSIAM J. Numer. Anal. 54 (2016), no. 6, 3388\u20133410.","DOI":"10.1137\/15M1032582"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_007","doi-asserted-by":"crossref","unstructured":"W. Dahmen, G. Kutyniok, W.-Q. Lim, C. Schwab and G. Welper,\nAdaptive anisotropic Petrov\u2013Galerkin methods for first order transport equations,\nJ. Comput. Appl. Math. 340 (2018), 191\u2013220.","DOI":"10.1016\/j.cam.2018.02.023"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_008","doi-asserted-by":"crossref","unstructured":"W. Dahmen and R. P. Stevenson,\nAdaptive strategies for transport equations,\nComput. Methods Appl. Math. 19 (2019), no. 3, 431\u2013464.","DOI":"10.1515\/cmam-2018-0230"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_009","unstructured":"L. Demkowicz,\nComputing with \n                  \n                     \n                        \n                           \n                              h\n                              \u2062\n                              p\n                           \n                        \n                        \n                        hp\n                     \n                  \n                Finite Elements. I. One- and Two-Dimensional Elliptic and Maxwell Problems,\nChapman & Hall\/CRC, Boca Raton, 2006."},{"key":"2023033112515576898_j_cmam-2021-0149_ref_010","doi-asserted-by":"crossref","unstructured":"H. De Sterck, T. A. Manteuffel, S. F. McCormick and L. Olson,\nLeast-squares finite element methods and algebraic multigrid solvers for linear hyperbolic PDEs,\nSIAM J. Sci. Comput. 26 (2004), no. 1, 31\u201354.","DOI":"10.1137\/S106482750240858X"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_011","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":"2023033112515576898_j_cmam-2021-0149_ref_012","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":"2023033112515576898_j_cmam-2021-0149_ref_013","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Gopalakrishnan, S. Nagaraj and P. Sep\u00falveda,\nA spacetime DPG method for the Schr\u00f6dinger equation,\nSIAM J. Numer. Anal. 55 (2017), no. 4, 1740\u20131759.","DOI":"10.1137\/16M1099765"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_014","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":"2023033112515576898_j_cmam-2021-0149_ref_015","doi-asserted-by":"crossref","unstructured":"L. Demkowicz and P. Zanotti,\nConstruction of DPG Fortin operators revisited,\nComput. Math. 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Sep\u00falveda,\nA tent pitching scheme motivated by Friedrichs theory,\nComput. Math. Appl. 70 (2015), no. 5, 1114\u20131135.","DOI":"10.1016\/j.camwa.2015.07.001"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_020","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":"2023033112515576898_j_cmam-2021-0149_ref_021","unstructured":"M. Jensen,\nDiscontinuous Galerkin methods for Friedrichs systems with irregular solutions,\nPhD thesis, Corpus Christi College, University of Oxford, 2004."},{"key":"2023033112515576898_j_cmam-2021-0149_ref_022","doi-asserted-by":"crossref","unstructured":"P. Joly,\nSome trace theorems in anisotropic Sobolev spaces,\nSIAM J. Math. Anal. 23 (1992), no. 3, 799\u2013819.","DOI":"10.1137\/0523042"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_023","doi-asserted-by":"crossref","unstructured":"J. Mu\u00f1oz Matute, D. Pardo and L. Demkowicz,\nA DPG-based time-marching scheme for linear hyperbolic problems,\nComput. Methods Appl. Mech. Engrg. 373 (2021), Paper No. 113539.","DOI":"10.1016\/j.cma.2020.113539"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_024","doi-asserted-by":"crossref","unstructured":"J. Mu\u00f1oz Matute, D. Pardo and L. Demkowicz,\nEquivalence between the DPG method and the exponential integrators for linear parabolic problems,\nJ. Comput. Phys. 429 (2021), Paper No. 110016.","DOI":"10.1016\/j.jcp.2020.110016"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_025","doi-asserted-by":"crossref","unstructured":"S. Nagaraj, S. Petrides and L. F. Demkowicz,\nConstruction of DPG Fortin operators for second order problems,\nComput. Math. Appl. 74 (2017), no. 8, 1964\u20131980.","DOI":"10.1016\/j.camwa.2017.05.030"},{"key":"2023033112515576898_j_cmam-2021-0149_ref_026","unstructured":"J. T. Oden and L. F. Demkowicz,\nApplied Functional Analysis, 3rd ed.,\nCRC Press, Boca Raton, 2018."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0149\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:16:22Z","timestamp":1680282982000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0149\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,22]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,10,6]]},"published-print":{"date-parts":[[2023,1,1]]}},"alternative-id":["10.1515\/cmam-2021-0149"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2021-0149","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2022,6,22]]}}}