椭圆型方程的迭代解

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-01-01 DOI:10.14232/ejqtde.2022.1.34

P. Korman, D. Schmidt

{"title":"椭圆型方程的迭代解","authors":"P. Korman, D. Schmidt","doi":"10.14232/ejqtde.2022.1.34","DOIUrl":null,"url":null,"abstract":"We reduce solution of the Dirichlet problem (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:math>) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mspace width=\"1em\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\"1em\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> to iterative solution of a simpler problem <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mspace width=\"thickmathspace\" /> <mml:mspace width=\"thickmathspace\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\"thickmathspace\" /> <mml:mspace width=\"thickmathspace\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\"thickmathspace\" /> <mml:mspace width=\"thickmathspace\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mspace width=\"thinmathspace\" /> <mml:mo>,</mml:mo> </mml:math> for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\"1em\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\"1em\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> </mml:math>.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Iterative solution of elliptic equations\",\"authors\":\"P. Korman, D. Schmidt\",\"doi\":\"10.14232/ejqtde.2022.1.34\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We reduce solution of the Dirichlet problem (<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:math>) <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <mml:mi mathvariant=\\\"normal\\\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mspace width=\\\"1em\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\\\"2em\\\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\\\"1em\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> to iterative solution of a simpler problem <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <mml:mi mathvariant=\\\"normal\\\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mspace width=\\\"thickmathspace\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mspace width=\\\"thinmathspace\\\" /> <mml:mo>,</mml:mo> </mml:math> for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <mml:mi mathvariant=\\\"normal\\\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\\\"1em\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>in </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=\\\"2em\\\" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=\\\"1em\\\" /> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mtext>on </mml:mtext> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> </mml:math>.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.14232/ejqtde.2022.1.34\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2022.1.34","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

摘要

我们将Dirichlet问题(x∈D∧R m) Δ u (x) + a (x) u (x) = f (x)在D中，u = 0在∂D中简化为一个更简单的问题Δ u = f (x)在D中，u = 0在∂D中，可以使用傅里叶级数或格林函数方法。该方法适用于数值计算，特别是当人们使用牛顿方法解决半线性问题Δ u + g (x, u) = 0在D中，u = 0在∂D中，。

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

Iterative solution of elliptic equations

We reduce solution of the Dirichlet problem ( x ∈ D ⊂ R m ) Δ u ( x ) + a ( x ) u ( x ) = f ( x ) in D , u = 0 on ∂ D to iterative solution of a simpler problem Δ u = f ( x ) in D , u = 0 on ∂ D , for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems Δ u + g ( x , u ) = 0 in D , u = 0 on ∂ D , .

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.