{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:32:39Z","timestamp":1787329959616,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1541585"],"award-info":[{"award-number":["DMS-1541585"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1719829"],"award-info":[{"award-number":["DMS-1719829"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1818861"],"award-info":[{"award-number":["DMS-1818861"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011132","name":"Rutgers, The State University of New Jersey","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100011132","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 develop discrete $W^2_p$-norm error estimates for the Oliker--Prussner method applied to the Monge--Amp\u00e8re equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $\\| u - u_h \\|_{W^2_{f,p}} (\\mathcal{N}^I_h)$ converges in order $O(h^{1\/p})$ if $p &gt; d$ and converges in order $O(h^{1\/d} \\ln (\\frac 1 h)^{1\/d})$ if $p \\leq d$, where $\\|\\cdot\\|_{W^2_{f,p}(\\mathcal{N}^I_h)}$ is a weighted $W^2_p$-type norm, and the constant $C&gt;0$ depends on $\\|{u}\\|_{C^{3,1}(\\bar\\Omega)}$, the dimension $d$, and the constant $p$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.<\/jats:p>","DOI":"10.1137\/17m1160409","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T13:29:31Z","timestamp":1539696571000},"page":"3099-3120","source":"Crossref","is-referenced-by-count":8,"title":["Rates of Convergence in $W^2_p$-Norm for the Monge--Amp\u00e8re Equation"],"prefix":"10.1137","volume":"56","author":[{"given":"Michael","family":"Neilan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wujun","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,10,16]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009187"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-014-0127-7"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-014-0524-y"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1991-4305"},{"key":"atypb5","volume-title":"Convergence Rates for Discretized Monge-Amp\u00e8re Equations and Quantitative Stability of Optimal Transport, arXiv:1803.00785 [math.NA]","author":"Berman R. 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