{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T08:52:20Z","timestamp":1781167940697,"version":"3.54.1"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"327","license":[{"start":{"date-parts":[[2021,7,27]],"date-time":"2021-07-27T00:00:00Z","timestamp":1627344000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913133"],"award-info":[{"award-number":["1913133"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We prove that the Galerkin finite element solution\n \n \n \n \n u<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n u_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the Laplace equation in a convex polyhedron\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\varOmega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree\n \n \n \n \n r<\/mml:mi>\n \n \u2a7e\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n r\\geqslant 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , satisfies the following weak maximum principle:\n \n \n \n \n \n \n \n \n \u2016<\/mml:mo>\n \n u<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \u2016<\/mml:mo>\n <\/mml:mrow>\n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2a7d\n \n <\/mml:mo>\n C<\/mml:mi>\n \n \n \u2016<\/mml:mo>\n \n u<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \u2016<\/mml:mo>\n <\/mml:mrow>\n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u2202\n \n <\/mml:mi>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n \\begin{align*} \\left \\|u_{h}\\right \\|_{L^{\\infty }(\\varOmega )} \\leqslant C\\left \\|u_{h}\\right \\|_{L^{\\infty }(\\partial \\varOmega )} , \\end{align*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n with a constant\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n independent of the mesh size\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . By using this result, we show that the Ritz projection operator\n \n \n \n \n R<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n R_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is stable in\n \n \n \n \n L<\/mml:mi>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msup>\n L^\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm uniformly in\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n r<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n r\\geq 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , i.e.,\n \n \n \n \n \n \n \n \u2016\n \n <\/mml:mo>\n \n R<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2a7d\n \n <\/mml:mo>\n C<\/mml:mi>\n \n \u2016\n \n <\/mml:mo>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n \\begin{align*} \\|R_hu\\|_{L^{\\infty }(\\varOmega )} \\leqslant C\\|u\\|_{L^{\\infty }(\\varOmega )} . \\end{align*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.\n <\/p>","DOI":"10.1090\/mcom\/3560","type":"journal-article","created":{"date-parts":[[2020,5,20]],"date-time":"2020-05-20T14:43:00Z","timestamp":1589985780000},"page":"1-18","source":"Crossref","is-referenced-by-count":15,"title":["Weak discrete maximum principle of finite element methods in convex polyhedra"],"prefix":"10.1090","volume":"90","author":[{"given":"Dmitriy","family":"Leykekhman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Buyang","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2020,7,27]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"1771","DOI":"10.1137\/080731724","article-title":"\ud835\udc3f^{\u221e}-error estimates on graded meshes with application to optimal control","volume":"48","author":"Apel, Thomas","year":"2009","journal-title":"SIAM J. 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