{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:49:48Z","timestamp":1787320188926,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>An $L^\\infty$-error estimate of the finite element approximation of an elliptic boundary value problem with Dirichlet boundary conditions in domains with corners is given. To achieve an approximation rate of $h^2|\\ln h|$ the mesh has to be appropriately graded near corners with an interior angle larger than $\\omega_0$, with $\\omega_0=\\frac{\\pi}{2}$ for the Poisson problem. In contrast to previous publications, the norm of the function that has to be approximated is separated from the constants in this estimate. This result is applied to a linear-quadratic optimal control problem with constraints on the control. Two approaches are considered, one where the control is approximated by piecewise constant functions and improved by a postprocessing step, and the other where the control is not discretized. For both approaches a convergence rate of $h^2|\\ln h|$ in the maximum norm is shown.<\/jats:p>","DOI":"10.1137\/080731724","type":"journal-article","created":{"date-parts":[[2009,5,20]],"date-time":"2009-05-20T18:27:35Z","timestamp":1242844055000},"page":"1771-1796","source":"Crossref","is-referenced-by-count":21,"title":["$L^\\infty$-Error Estimates on Graded Meshes with Application to Optimal Control"],"prefix":"10.1137","volume":"48","author":[{"given":"Thomas","family":"Apel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arnd","family":"R\u00f6sch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dieter","family":"Sirch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,5,20]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-007-0133-0"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01399326"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02567822"},{"key":"R5","unstructured":"J. Frehse and R. Rannacher,\n                      Eine ${L}^1$-Fehlerabsch\u00e4tzung f\u00fcr diskrete Grundl\u00f6sungen in der Methode der finiten Elemente\n                      , in Finite Elemente. Tagungsband des Sonderforschungsbereichs 72, J. Frehse, R. Leis, and R. Schaback, eds., Bonner Mathematische Schriften 89, Bonn, 1976, pp. 92\u2013114."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-005-4559-5"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-003-0120-1"},{"key":"R8","first-page":"209","volume":"16","author":"Kondrat'ev V. A.","year":"1967","journal-title":"Trudy Moskov. Mat. Obshch."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"V. A. Kozlov, V. G. Maz'ya, and J. 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Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012903431608"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/040614621"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01419529"},{"key":"R16","unstructured":"L. A. Oganesyan and L. A. Rukhovets,\n                      Variational-difference methods for the solution of elliptic equations\n                      , Izd. Akad. Nauk Armyanskoi SSR, Jerevan, 1979 (in Russian)."},{"key":"R17","unstructured":"L. A. Oganesyan, L. Rukhovets, and V. Rivkind,\n                      Variational-difference methods for solving elliptic equations, Part\n                      II, vol. 8 of Differential equations and their applications, Izd. Akad. Nauk Lit. SSR, Vilnius, 1974 (in Russian)."},{"key":"R18","unstructured":"G. Raugel,\n                      R\u00e9solution num\u00e9rique de probl\u00e8mes elliptiques dans des domaines avec coins\n                      , Ph.D. thesis, Universit\u00e9 de Rennes, 1978."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/388"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431753-X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1978-0502065-1"},{"key":"R22","first-page":"465","volume":"33","author":"Schatz A. H.","year":"1979","journal-title":"Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436386"},{"key":"R24","first-page":"681","volume":"30","author":"Scott R.","year":"1976","journal-title":"Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R25","unstructured":"J. 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