{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:16:42Z","timestamp":1760267802522,"version":"3.40.5"},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems.\nOptimal error estimates are shown in a natural norm and in the <jats:inline-formula id=\"j_cmam-2021-0056_ineq_9999\">\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-2021-0056_eq_0259.png\"\/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> norm of the field variable.\nFor the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature.\nIn the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators.\nIt is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.<\/jats:p>","DOI":"10.1515\/cmam-2021-0056","type":"journal-article","created":{"date-parts":[[2021,7,16]],"date-time":"2021-07-16T22:34:23Z","timestamp":1626474863000},"page":"811-826","source":"Crossref","is-referenced-by-count":4,"title":["Analysis of Backward Euler Primal DPG Methods"],"prefix":"10.1515","volume":"21","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5171-1457","authenticated-orcid":false,"given":"Norbert","family":"Heuer","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , 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 , Valpara\u00edso , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,7,17]]},"reference":[{"key":"2023033111295710430_j_cmam-2021-0056_ref_001","doi-asserted-by":"crossref","unstructured":"R.  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