{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:29:01Z","timestamp":1787329741595,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1411808"],"award-info":[{"award-number":["DMS-1411808"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We derive optimal error estimates in the max norm for the two-scale method for the Monge--Amp\u00e8re equation introduced in [R. H. Nochetto, D. Ntogkas, and W. Zhang, \u201cTwo-scale method for the Monge-Amp\u00e8re equation: Convergence to the viscosity solution,\u201d Math. Comp., 2018], [R. H. Nochetto, D. Ntogkas, and W. Zhang, \u201cTwo-scale method for the Monge-Amp\u00e8re equation: Pointwise convergence rates,\u201d IMA J. Numer. Anal., 2018, dry026] for dimension $d\\ge 2$. The technique hinges on the discrete comparison principle and the construction of suitable discrete barrier functions and applies to classical solutions with either H\u00f6lder or Sobolev regularity. We also explore connections with the semi-Lagrangian method of [X. Feng and M. Jensen, SIAM J. Numer. Anal., 55 (2017), pp. 691--712] and establish optimal pointwise error estimates as well.<\/jats:p>","DOI":"10.1137\/18m1165670","type":"journal-article","created":{"date-parts":[[2018,6,28]],"date-time":"2018-06-28T12:53:20Z","timestamp":1530190400000},"page":"1915-1941","source":"Crossref","is-referenced-by-count":7,"title":["Optimal Pointwise Error Estimates for Two-Scale Methods for the Monge--Amp\u00e8re Equation"],"prefix":"10.1137","volume":"56","author":[{"given":"Wenbo","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ricardo H.","family":"Nochetto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dru028"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3080"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/040621740"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2011-02487-7"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2011067"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160370306"},{"key":"atypb8","first-page":"71","volume":"22","author":"Dean E. 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