{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T19:45:23Z","timestamp":1766087123197},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We investigate a weak space-time formulation of the\nheat equation and its use for the construction of a numerical scheme.\nThe formulation is based on a known weak space-time formulation, with\nthe difference that a pointwise component of the solution, which in\nother works is usually neglected, is now kept. We investigate the role\nof such a component by first using it to obtain a pointwise bound on\nthe solution and then deploying it to construct a numerical scheme.\nThe scheme obtained, besides being quasi-optimal in the <jats:inline-formula id=\"j_cmam-2016-0027_ineq_9999_w2aab3b7d498b1b6b1aab1c13b1b1Aa\">\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:tex-math>${L^{2}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> sense, is also\npointwise superconvergent in the temporal nodes. We prove <jats:italic>a priori<\/jats:italic> error estimates and we present numerical experiments to empirically support our findings.<\/jats:p>","DOI":"10.1515\/cmam-2016-0027","type":"journal-article","created":{"date-parts":[[2016,9,27]],"date-time":"2016-09-27T09:16:43Z","timestamp":1474967803000},"page":"65-84","source":"Crossref","is-referenced-by-count":13,"title":["Numerical Solution of Parabolic Problems Based on a Weak Space-Time Formulation"],"prefix":"10.1515","volume":"17","author":[{"given":"Stig","family":"Larsson","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE\u201341296 Gothenburg, Sweden"}]},{"given":"Matteo","family":"Molteni","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Chalmers University of Technologyand University of Gothenburg, SE\u201341296 Gothenburg, Sweden"}]}],"member":"374","published-online":{"date-parts":[[2016,9,27]]},"reference":[{"key":"2023033115185148477_j_cmam-2016-0027_ref_001_w2aab3b7d498b1b6b1ab2ab1Aa","unstructured":"Andreev R.,\nStability of space-time Petrov\u2013Galerkin discretizations for parabolic evolution equations,\nPhD thesis, Dissertation no. 20842, ETH Z\u00fcrich, 2012."},{"key":"2023033115185148477_j_cmam-2016-0027_ref_002_w2aab3b7d498b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Andreev R.,\nStability of sparse space-time finite element discretizations of linear parabolic evolution equations,\nIMA J. Numer. Anal. 33 (2013), no. 1, 242\u2013260.","DOI":"10.1093\/imanum\/drs014"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_003_w2aab3b7d498b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"Andreev R.,\nOn long time integration of the heat equation,\nCalcolo 53 (2016), no. 1, 19\u201334.","DOI":"10.1007\/s10092-014-0133-9"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_004_w2aab3b7d498b1b6b1ab2ab4Aa","unstructured":"Babu\u0161ka I. and Aziz A. K.,\nSurvey lectures on the mathematical foundations of the finite element method,\nThe Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Baltimore 1972),\nAcademic Press, New York (1972), 1\u2013359."},{"key":"2023033115185148477_j_cmam-2016-0027_ref_005_w2aab3b7d498b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka I. and Janik T.,\nThe h-p version of the finite element method for parabolic equations. I. The p-version in time,\nNumer. Methods Partial Differential Equations 5 (1989), no. 4, 363\u2013399.","DOI":"10.1002\/num.1690050407"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_006_w2aab3b7d498b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka I. and Janik T.,\nThe h-p version of the finite element method for parabolic equations. II. The h-p version in time,\nNumer. Methods Partial Differential Equations 6 (1990), no. 4, 343\u2013369.","DOI":"10.1002\/num.1690060406"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_007_w2aab3b7d498b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"N. Chegini  and R. Stevenson ,\nAdaptive wavelet schemes for parabolic problems: Sparse matrices and numerical results,\nSIAM J. Numer. Anal. 49 (2011), no. 1, 182\u2013212.","DOI":"10.1137\/100800555"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_008_w2aab3b7d498b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Cioica P. A., Dahlke S., D\u00f6hring N., Friedrich U., Kinzel S., Lindner F., Raasch T., Ritter K. and Schilling R. L.,\nConvergence analysis of spatially adaptive Rothe methods,\nFound. Comput. Math. 14 (2014), no. 5, 863\u2013912.","DOI":"10.1007\/s10208-013-9183-7"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_009_w2aab3b7d498b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Ern A. and Guermond J.,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"2023033115185148477_j_cmam-2016-0027_ref_010_w2aab3b7d498b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"Larsson S. and Molteni M.,\nA weak space-time formulation for the linear stochastic heat equation,\nInt. J. Appl. Comput. Math. 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