{"title":"有限体积离散的重叠域分解方法","authors":"Jinjin Zhang , Yanru Su , Xinfeng Gao , Xuemin Tu","doi":"10.1016/j.camwa.2024.10.018","DOIUrl":null,"url":null,"abstract":"<div><div>Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm, an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> norm, and an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Overlapping domain decomposition methods for finite volume discretizations\",\"authors\":\"Jinjin Zhang , Yanru Su , Xinfeng Gao , Xuemin Tu\",\"doi\":\"10.1016/j.camwa.2024.10.018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm, an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> norm, and an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.</div></div>\",\"PeriodicalId\":55218,\"journal\":{\"name\":\"Computers & Mathematics with Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-10-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computers & Mathematics with Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0898122124004619\",\"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":"Computers & Mathematics with Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0898122124004619","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Overlapping domain decomposition methods for finite volume discretizations
Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete norm, an norm, and an semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.
期刊介绍:
Computers & Mathematics with Applications provides a medium of exchange for those engaged in fields contributing to building successful simulations for science and engineering using Partial Differential Equations (PDEs).