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\n We consider an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for a model of second-order elliptic problem on a convex polygonal or polyhedral domain. Our first main goal is to prove the best and the almost best approximation properties of the errors in the\n \n \n \n \n W<\/mml:mi>\n \n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n W^{1,\\infty }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n L^{\\infty }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norms by an interior error estimate introduced in this paper. The second main contribution is an application of the approximation property in the\n \n \n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n L^{\\infty }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm to the convergence analysis for IP-HDG methods for an optimal control problem with point values of the state in the objective functional. Since the low regularity induced by the Dirac function on the right-hand side of the adjoint state equation, the standard\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -projection and Lagrange interpolation can only get the convergence rates\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n d<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n \u03f5\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h^{2-d\/2-\\epsilon })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n d<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h^{2-d\/2})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for piecewise constant and variational discretizations for the\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm of the control, where\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a positive constant that can be arbitrarily small. However, with the help of the approximation property in the\n \n \n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n L^{\\infty }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm, the convergence rate for\n \n \n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n d=3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n can be improved to be\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n h<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n h<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h|\\ln h|)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Finally, two numerical examples in two dimensions are provided to validate the theoretical analysis.\n <\/p>","DOI":"10.1090\/mcom\/3923","type":"journal-article","created":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T09:52:37Z","timestamp":1697622757000},"page":"2155-2184","source":"Crossref","is-referenced-by-count":1,"title":["\ud835\udc4a^{1,\u221e} and \ud835\udc3f^{\u221e} error estimates for an IP-HDG method for second-order elliptic problem with application to pointwise tracking optimal control"],"prefix":"10.1090","volume":"95","author":[{"given":"Haitao","family":"Leng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huangxin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2026,4,7]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"852","DOI":"10.1093\/imanum\/drx018","article-title":"Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems","volume":"38","author":"Antil, Harbir","year":"2018","journal-title":"IMA J. 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Wang, Finite Element Methods, Vol. 58 of Series in Information and Computation Science, Science Press, Beijing, 2013."},{"key":"39","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/112","volume-title":"Optimal control of partial differential equations","volume":"112","author":"Tr\u00f6ltzsch, Fredi","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821849040"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2026-03923-7\/mcom3923_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2026-03923-7\/S0025-5718-2026-03923-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:53:08Z","timestamp":1781286788000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2026-03923-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,7]]},"references-count":39,"journal-issue":{"issue":"361","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["S0025-5718-2026-03923-7"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3923","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":[[2026,4,7]]}}}