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Math."],"published-print":{"date-parts":[[2024,2]]},"abstract":"Abstract<\/jats:title>We consider the first-order system space\u2013time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by F\u00fchrer and Karkulik (Comput Math Appl 92:27\u201336, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283\u2013299 2021), with solution components $$(u_1,\\textbf{u}_2)=(u,-\\nabla _\\textbf{x} u)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \n u<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n (<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n -<\/mml:mo>\n \n \u2207<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The corresponding operator is boundedly invertible between a Hilbert space U<\/jats:italic> and a Cartesian product of $$L_2$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides $$L_2$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-norms of $$\\nabla _\\textbf{x} u_1$$<\/jats:tex-math>\n \n \n \u2207<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n \n u<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textbf{u}_2$$<\/jats:tex-math>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the (graph) norm of U<\/jats:italic> contains the $$L_2$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-norm of $$\\partial _t u_1 +{{\\,\\textrm{div}\\,}}_\\textbf{x} \\textbf{u}_2$$<\/jats:tex-math>\n \n \n \u2202<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n \n u<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \n div<\/mml:mtext>\n \n <\/mml:mrow>\n x<\/mml:mi>\n <\/mml:msub>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. When applying standard finite elements w.r.t.\u00a0simplicial partitions of the space\u2013time cylinder, estimates of the approximation error w.r.t.\u00a0the latter norm require higher-order smoothness of $$\\textbf{u}_2$$<\/jats:tex-math>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u<\/jats:italic>. In this paper, we construct finite element spaces w.r.t.\u00a0prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of $$\\partial _t u_1 +{{\\,\\textrm{div}\\,}}_\\textbf{x} \\textbf{u}_2$$<\/jats:tex-math>\n \n \n \u2202<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n \n u<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \n div<\/mml:mtext>\n \n <\/mml:mrow>\n x<\/mml:mi>\n <\/mml:msub>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, i.e., of the forcing term $$f=(\\partial _t-\\Delta _x)u$$<\/jats:tex-math>\n \n f<\/mml:mi>\n =<\/mml:mo>\n (<\/mml:mo>\n \n \u2202<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n -<\/mml:mo>\n \n \u0394<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Numerical results show significantly improved convergence rates.<\/jats:p>","DOI":"10.1007\/s00211-023-01387-3","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T23:02:24Z","timestamp":1703718144000},"page":"133-157","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Improved rates for a space\u2013time FOSLS of parabolic PDEs"],"prefix":"10.1007","volume":"156","author":[{"given":"Gregor","family":"Gantner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rob","family":"Stevenson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"issue":"1","key":"1387_CR1","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1093\/imanum\/drs014","volume":"33","author":"R Andreev","year":"2013","unstructured":"Andreev, R.: Stability of sparse space\u2013time finite element discretizations of linear parabolic evolution equations. 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