{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T00:29:44Z","timestamp":1778718584352,"version":"3.51.4"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"Abstract<\/jats:title>\n We present and analyze a space-time Petrov\u2013Galerkin finite element method for a time-fractional diffusion equation\ninvolving a Riemann\u2013Liouville fractional derivative of order \n \n \n \n \u03b1<\/m:mi>\n \u2208<\/m:mo>\n \n (<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n 1<\/m:mn>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\alpha\\in(0,1)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in time and zero initial data.\nWe derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness.\nFurther, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and\nderive error bounds in both energy and \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norms for the finite element solution. In the proof\nof the discrete inf-sup condition, a certain nonstandard \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> stability property of the \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> projection operator plays a key role.\nWe provide extensive numerical examples to verify the convergence analysis.<\/jats:p>","DOI":"10.1515\/cmam-2017-0026","type":"journal-article","created":{"date-parts":[[2017,8,26]],"date-time":"2017-08-26T10:00:40Z","timestamp":1503741640000},"page":"1-20","source":"Crossref","is-referenced-by-count":19,"title":["Space-Time Petrov\u2013Galerkin FEM for Fractional Diffusion Problems"],"prefix":"10.1515","volume":"18","author":[{"given":"Beiping","family":"Duan","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics , Central South University , 410083 Changsha , P. R. China ; and Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bangti","family":"Jin","sequence":"additional","affiliation":[{"name":"Department of Computer Science , University College London , Gower Street , London WC1E 6BT , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raytcho","family":"Lazarov","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Texas A&M University , College Station , TX 77843-3368 , USA ; and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Pasciak","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Texas A&M University , College Station , TX 77843-3368 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics , The Hong Kong Polytechnic University , Kowloon , Hong Kong , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,8,26]]},"reference":[{"key":"2023033115122817085_j_cmam-2017-0026_ref_001_w2aab3b7d487b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"O. Axelsson and J. Maubach,\nA time-space finite element discretization technique for the calculation of the electromagnetic field in ferromagnetic materials,\nInternat. J. Numer. 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Math. 15 (2015), no. 4, 551\u2013566.","DOI":"10.1515\/cmam-2015-0026"},{"key":"2023033115122817085_j_cmam-2017-0026_ref_029_w2aab3b7d487b1b6b1ab2ac29Aa","unstructured":"V. Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2023033115122817085_j_cmam-2017-0026_ref_030_w2aab3b7d487b1b6b1ab2ac30Aa","doi-asserted-by":"crossref","unstructured":"M. Zayernouri, M. Ainsworth and G. E. Karniadakis,\nA unified Petrov\u2013Galerkin spectral method for fractional PDEs,\nComput. Methods Appl. Mech. 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