{"title":"蒙日-安培方程近似值的稳定性和保证误差控制","authors":"Dietmar Gallistl, Ngoc Tien Tran","doi":"10.1007/s00211-023-01385-5","DOIUrl":null,"url":null,"abstract":"<p>This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the <span>\\(L^\\infty \\)</span> norm from the theory of viscosity solutions which are independent of the regularization parameter <span>\\(\\varepsilon \\)</span>. They allow for the uniform convergence of the solution <span>\\(u_\\varepsilon \\)</span> to the regularized problem towards the Alexandrov solution <i>u</i> to the Monge–Ampère equation for any nonnegative <span>\\(L^n\\)</span> right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the <span>\\(L^\\infty \\)</span> norm for continuously differentiable finite element approximations of <i>u</i> or <span>\\(u_\\varepsilon \\)</span>.\n</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stability and guaranteed error control of approximations to the Monge–Ampère equation\",\"authors\":\"Dietmar Gallistl, Ngoc Tien Tran\",\"doi\":\"10.1007/s00211-023-01385-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the <span>\\\\(L^\\\\infty \\\\)</span> norm from the theory of viscosity solutions which are independent of the regularization parameter <span>\\\\(\\\\varepsilon \\\\)</span>. They allow for the uniform convergence of the solution <span>\\\\(u_\\\\varepsilon \\\\)</span> to the regularized problem towards the Alexandrov solution <i>u</i> to the Monge–Ampère equation for any nonnegative <span>\\\\(L^n\\\\)</span> right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the <span>\\\\(L^\\\\infty \\\\)</span> norm for continuously differentiable finite element approximations of <i>u</i> or <span>\\\\(u_\\\\varepsilon \\\\)</span>.\\n</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2023-12-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00211-023-01385-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-023-01385-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
Stability and guaranteed error control of approximations to the Monge–Ampère equation
This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the \(L^\infty \) norm from the theory of viscosity solutions which are independent of the regularization parameter \(\varepsilon \). They allow for the uniform convergence of the solution \(u_\varepsilon \) to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative \(L^n\) right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the \(L^\infty \) norm for continuously differentiable finite element approximations of u or \(u_\varepsilon \).