{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T19:46:02Z","timestamp":1777751162767,"version":"3.51.4"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"281","license":[{"start":{"date-parts":[[2013,8,8]],"date-time":"2013-08-08T00:00:00Z","timestamp":1375920000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper we study the a priori error estimates for the finite element approximations of parabolic equations with measure data, especially we consider problems with separate measure data in time and space, respectively. The solutions of these kinds of problems exhibit low regularities due to the existence of measure data, this introduces some difficulties in both theoretical and numerical analysis. For both cases we use standard piecewise linear and continuous finite elements for the space discretization and derive the a priori error estimates for the semi-discretization problems, while the backward Euler method is then used for time discretization and a priori error estimates for the fully discrete problems are also derived. Numerical results are provided at the end of the paper to confirm our theoretical findings.<\/p>","DOI":"10.1090\/s0025-5718-2012-02630-5","type":"journal-article","created":{"date-parts":[[2012,8,8]],"date-time":"2012-08-08T12:30:16Z","timestamp":1344429016000},"page":"69-98","source":"Crossref","is-referenced-by-count":21,"title":["Error estimates for finite element approximations of parabolic equations with measure data"],"prefix":"10.1090","volume":"82","author":[{"given":"Wei","family":"Gong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2012,8,8]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s00211-006-0041-2","article-title":"A posteriori error estimates for elliptic problems with Dirac delta source terms","volume":"105","author":"Araya, Rodolfo","year":"2006","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1006\/jfan.1996.3040","article-title":"Nonlinear parabolic equations with measure data","volume":"147","author":"Boccardo, Lucio","year":"1997","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"L. Boccardo and T. Gallou\u00ebt, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), pp. 149-169.","DOI":"10.1016\/0022-1236(89)90005-0"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1007\/BF01389461","article-title":"\ud835\udc3f\u00b2 estimates for the finite element method for the Dirichlet problem with singular data","volume":"47","author":"Casas, Eduardo","year":"1985","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1137\/S0363012995283637","article-title":"Pontryagin\u2019s principle for state-constrained boundary control problems of semilinear parabolic equations","volume":"35","author":"Casas, Eduardo","year":"1997","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1137\/S0036142900377991","article-title":"Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions","volume":"40","author":"Chrysafinos, K.","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet, The finite element methods for elliptic problems, North-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"issue":"5","key":"9","doi-asserted-by":"publisher","first-page":"1937","DOI":"10.1137\/060652361","article-title":"Convergence of a finite element approximation to a state-constrained elliptic control problem","volume":"45","author":"Deckelnick, Klaus","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4208\/jcm.1006-m3213","article-title":"Variational discretization of parabolic control problems in the presence of pointwise state constraints","volume":"29","author":"Deckelnick, Klaus","year":"2011","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/S0362-546X(98)00170-9","article-title":"Optimal pointwise control of semilinear parabolic equations","volume":"39","author":"Droniou, J.","year":"2000","journal-title":"Nonlinear Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-546X","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1137\/0732033","article-title":"Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \ud835\udc3f_{\u221e}\ud835\udc3f\u2082 and \ud835\udc3f_{\u221e}\ud835\udc3f_{\u221e}","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"201","key":"13","doi-asserted-by":"publisher","first-page":"79","DOI":"10.2307\/2153157","article-title":"Analysis of a robust finite element approximation for a parabolic equation with rough boundary data","volume":"60","author":"French, Donald A.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1017\/s0962492900002543","article-title":"Exact and approximate controllability for distributed parameter systems","author":"Glowinski, R.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521482550"},{"key":"15","doi-asserted-by":"crossref","unstructured":"W. Gong and M. Hinze, Error estimates for parabolic optimal control problems with control and state constraints, preprint, Hamburger Beitr\u00e4ge zur Angewandten Mathematik, 2010-13.","DOI":"10.1007\/s10589-013-9541-z"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1007\/s10915-010-9392-z","article-title":"A mixed finite element scheme for optimal control problems with pointwise state constraints","volume":"46","author":"Gong, Wei","year":"2011","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"17","unstructured":"R. Li and W. B. Liu, AFEPack available online at http:\/\/circus.math.pku.edu.cn\/AFEPack."},{"key":"18","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 170","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65024-6","volume-title":"Optimal control of systems governed by partial differential equations","author":"Lions, J.-L.","year":"1971"},{"key":"19","doi-asserted-by":"crossref","unstructured":"J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.","DOI":"10.1007\/978-3-642-65217-2"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1137\/0719003","article-title":"On the smoothing property of the Galerkin method for parabolic equations","volume":"19","author":"Luskin, Mitchell","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"21","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1137\/S0363012998345640","article-title":"Theoretical and numerical analysis of an optimal control problem related to wastewater treatment","volume":"38","author":"Mart\u00ednez, A.","year":"2000","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"5","key":"22","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1137\/100793888","article-title":"A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time","volume":"49","author":"Meidner, Dominik","year":"2011","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1137\/S0036142994269526","article-title":"Convergence past singularities for a fully discrete approximation of curvature-driven interfaces","volume":"34","author":"Nochetto, Ricardo H.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"24","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical approximation of partial differential equations","volume":"23","author":"Quarteroni, Alfio","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540571116"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1023\/A:1017907930931","article-title":"Pointwise control of the Burgers equation and related Nash equilibrium problems: computational approach","volume":"112","author":"Ramos, A. M.","year":"2002","journal-title":"J. Optim. Theory Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-3239","issn-type":"print"},{"issue":"158","key":"26","doi-asserted-by":"publisher","first-page":"437","DOI":"10.2307\/2007280","article-title":"Some optimal error estimates for piecewise linear finite element approximations","volume":"38","author":"Rannacher, Rolf","year":"1982","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"223","key":"27","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1090\/S0025-5718-98-00959-4","article-title":"Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates","volume":"67","author":"Schatz, Alfred H.","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"28","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/BF01436386","article-title":"Finite element convergence for singular data","volume":"21","author":"Scott, Ridgway","year":"1973","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"136","key":"29","doi-asserted-by":"publisher","first-page":"681","DOI":"10.2307\/2005390","article-title":"Optimal \ud835\udc3f^{\u221e} estimates for the finite element method on irregular meshes","volume":"30","author":"Scott, Ridgway","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"30","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Galerkin finite element methods for parabolic problems","volume":"25","author":"Thom\u00e9e, Vidar","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540331216","edition":"2"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2013-82-281\/S0025-5718-2012-02630-5\/S0025-5718-2012-02630-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-281\/S0025-5718-2012-02630-5\/S0025-5718-2012-02630-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:24:34Z","timestamp":1776792274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-281\/S0025-5718-2012-02630-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,8,8]]},"references-count":30,"journal-issue":{"issue":"281","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["S0025-5718-2012-02630-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2012-02630-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2012,8,8]]}}}