{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T12:29:48Z","timestamp":1778588988212,"version":"3.51.4"},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We introduce and analyze a discontinuous Petrov\u2013Galerkin method with optimal test functions\nfor the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak\nvariational formulation at each time step. We prove the stability of the method for\nthe field variables (the original unknown and its gradient weighted by the square root\nof the time step) and derive a C\u00e9a-type error estimate. For low-order approximation spaces\nthis implies certain convergence orders when time steps are not too small in comparison with mesh sizes.\nSome numerical experiments are reported to support our theoretical results.<\/jats:p>","DOI":"10.1515\/cmam-2016-0037","type":"journal-article","created":{"date-parts":[[2016,11,22]],"date-time":"2016-11-22T10:01:02Z","timestamp":1479808862000},"page":"237-252","source":"Crossref","is-referenced-by-count":14,"title":["A Time-Stepping DPG Scheme for the Heat Equation"],"prefix":"10.1515","volume":"17","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"}]},{"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"}]},{"given":"Jhuma","family":"Sen Gupta","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Pontificia Universidad Cat\u00f3lica de Chile, Avenida Vicu\u00f1a Mackenna 4860, Santiago, Chile"}]}],"member":"374","published-online":{"date-parts":[[2016,11,22]]},"reference":[{"key":"2023033114222311880_j_cmam-2016-0037_ref_001_w2aab3b7b1b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Bochev P. B., Gunzburger M. D. and Shadid J. N.,\nOn inf-sup stabilized finite element methods for transient problems,\nComput. Methods Appl. Mech. Engrg. 193 (2004), 1471\u20131489.","DOI":"10.1016\/j.cma.2003.12.034"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_002_w2aab3b7b1b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Bornemann F. A.,\nAn adaptive multilevel approach to parabolic equations,\nImpact Comput. Sci. Engrg. 2 (1990), 279\u2013317.","DOI":"10.1016\/0899-8248(90)90016-4"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_003_w2aab3b7b1b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"Bramble J. H. and Thom\u00e9e V.,\nSemidiscrete-least squares methods for parabolic boundary value problem,\nMath. Comp. 26 (1972), no. 119, 633\u2013648.","DOI":"10.1090\/S0025-5718-1972-0349038-4"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_004_w2aab3b7b1b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"Broersen D. and Stevenson R.,\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":"2023033114222311880_j_cmam-2016-0037_ref_005_w2aab3b7b1b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"Carstensen C., Demkowicz L. and Gopalakrishnan J.,\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":"2023033114222311880_j_cmam-2016-0037_ref_006_w2aab3b7b1b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"Chan J., Heuer N., Bui-Thanh T. and Demkowicz L.,\nRobust 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":"2023033114222311880_j_cmam-2016-0037_ref_007_w2aab3b7b1b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"Demkowicz L. and Gopalakrishnan J.,\nA class of discontinuous Petrov\u2013Galerkin methods. Part II: Optimal test functions,\nNumer. Methods Partial Differential Equations 27 (2011), 70\u2013105.","DOI":"10.1002\/num.20640"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_008_w2aab3b7b1b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Demkowicz L. and Gopalakrishnan J.,\nAnalysis of the DPG method for the Poisson problem,\nSIAM J. Numer. Anal. 49 (2011), no. 5, 1788\u20131809.","DOI":"10.1137\/100809799"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_009_w2aab3b7b1b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Demkowicz L. and Heuer N.,\nRobust DPG method for convection-dominated diffusion problems,\nSIAM J. Numer. Anal. 51 (2013), no. 5, 2514\u20132537.","DOI":"10.1137\/120862065"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_010_w2aab3b7b1b1b6b1ab2ac10Aa","unstructured":"Ellis T., Chan J. and Demkowicz L.,\nRobust DPG method for transient convection-diffusion,\nICES Report 15-21, The University of Texas at Austin, 2015."},{"key":"2023033114222311880_j_cmam-2016-0037_ref_011_w2aab3b7b1b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"F\u00fchrer T. and Heuer N.,\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":"2023033114222311880_j_cmam-2016-0037_ref_012_w2aab3b7b1b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"Gopalakrishnan J. and Qiu W.,\nAn analysis of the practical DPG method,\nMath. Comp. 83 (2014), no. 286, 537\u2013552.","DOI":"10.1090\/S0025-5718-2013-02721-4"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_013_w2aab3b7b1b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"Harari I.,\nStability of semidiscrete formulations for parabolic problems at small time steps,\nComput. Methods Appl. Mech. Engrg. 193 (2004), 1491\u20131516.","DOI":"10.1016\/j.cma.2003.12.035"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_014_w2aab3b7b1b1b6b1ab2ac14Aa","unstructured":"Heuer N. and Karkulik M.,\nA robust DPG method for singularly perturbed reaction-diffusion problems,\npreprint 2015, https:\/\/arxiv.org\/abs\/1509.07560."},{"key":"2023033114222311880_j_cmam-2016-0037_ref_015_w2aab3b7b1b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"Majidi M. and Starke G.,\nLeast-Squares Galerkin methods for parabolic problems I: Semidiscretization in time,\nSIAM J. Numer. Anal. 39 (2001), no. 4, 1302\u20131323.","DOI":"10.1137\/S0036142900370125"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_016_w2aab3b7b1b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"Majidi M. and Starke G.,\nLeast-Squares Galerkin methods for parabolic problems II: The fully discrete case and adaptive algorithms,\nSIAM J. Numer. Anal. 39 (2002), no. 5, 1648\u20131666.","DOI":"10.1137\/S0036142900379461"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_017_w2aab3b7b1b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"Niemi A. H., Collier N. O. and Calo V. M.,\nAutomatically stable discontinuous Petrov\u2013Galerkin methods for stationary transport problems: Quasi-optimal test space norm,\nComput. Math. Appl. 66 (2013), no. 10, 2096\u20132113.","DOI":"10.1016\/j.camwa.2013.07.016"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_018_w2aab3b7b1b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"Rothe E.,\nZweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,\nMath. Ann. 102 (1930), no. 1, 650\u2013670.","DOI":"10.1007\/BF01782368"},{"key":"2023033114222311880_j_cmam-2016-0037_ref_019_w2aab3b7b1b1b6b1ab2ac19Aa","unstructured":"Thom\u00e9e V.,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2023033114222311880_j_cmam-2016-0037_ref_020_w2aab3b7b1b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"Zitelli J., Muga I., Demkowicz L., Gopalakrishnan J., Pardo D. and Calo V. M.,\nA class of discontinuous Petrov\u2013Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D,\nJ. Comput. Phys. 230 (2011), no. 7, 2406\u20132432.","DOI":"10.1016\/j.jcp.2010.12.001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/2\/article-p237.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0037\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0037\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T20:39:17Z","timestamp":1680295157000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0037\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,22]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,1,10]]},"published-print":{"date-parts":[[2017,4,1]]}},"alternative-id":["10.1515\/cmam-2016-0037"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2016-0037","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2016,11,22]]}}}