混合边界条件下椭圆型问题的低正则性估计

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2023-08-07 DOI:10.1090/mcom/3875

Erik Burman, Peter Hansbo, Mats G. Larson

{"title":"混合边界条件下椭圆型问题的低正则性估计","authors":"Erik Burman, Peter Hansbo, Mats G. Larson","doi":"10.1090/mcom/3875","DOIUrl":null,"url":null,"abstract":"We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u element-of upper H Superscript s\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u \\in H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s element-of left-parenthesis 1 comma 3 slash 2 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">s\\in (1,3/2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"33 1","pages":"0"},"PeriodicalIF":2.2000,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions\",\"authors\":\"Erik Burman, Peter Hansbo, Mats G. Larson\",\"doi\":\"10.1090/mcom/3875\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"u element-of upper H Superscript s\\\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">u \\\\in H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"s element-of left-parenthesis 1 comma 3 slash 2 right-bracket\\\"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=\\\"false\\\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">s\\\\in (1,3/2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.\",\"PeriodicalId\":18456,\"journal\":{\"name\":\"Mathematics of Computation\",\"volume\":\"33 1\",\"pages\":\"0\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2023-08-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics of Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3875\",\"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":"Mathematics of Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mcom/3875","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

给出了具有混合边界条件的二阶椭圆型问题的切割有限元近似的误差估计。误差估计是低正则型的，我们考虑当精确解u∈H s u \in H^s与s∈(1,3/2)s\in(1,3/2)的情况。对于Nitsche型方法，这种情况需要对涉及精确解在边界处的法向通量的项进行特殊处理。对于狄利克雷边界条件，估计是最优的，而在混合狄利克雷-诺伊曼边界条件的情况下，它们是次优的对数因子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ H s u \in H^s with s ∈ ( 1 , 3 / 2 ] s\in (1,3/2] . For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.