{"title":"基于多级离散化的Steklov特征值问题局部缺陷校正方法","authors":"Fei Xu, Liu Chen, Qiumei Huang","doi":"10.1051/m2an/2021076","DOIUrl":null,"url":null,"abstract":"In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem\",\"authors\":\"Fei Xu, Liu Chen, Qiumei Huang\",\"doi\":\"10.1051/m2an/2021076\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.\",\"PeriodicalId\":50499,\"journal\":{\"name\":\"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2021-11-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1051/m2an/2021076\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/m2an/2021076","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem
In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.
期刊介绍:
M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem.
Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.