{"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":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1000,"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\":49733,\"journal\":{\"name\":\"Numerische Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-12-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerische Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00211-023-01385-5\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerische Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-023-01385-5","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","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 \(u_\varepsilon \).
期刊介绍:
Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers:
1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis)
2. Optimization and Control Theory
3. Mathematical Modeling
4. The mathematical aspects of Scientific Computing