Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler
{"title":"半线性椭圆型单反应边值问题hp-ILGFEM的指数收敛性","authors":"Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler","doi":"10.1093/imanum/draf030","DOIUrl":null,"url":null,"abstract":"We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\\varOmega \\subset{\\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\\text{H}^{1}(\\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exponential convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction\",\"authors\":\"Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler\",\"doi\":\"10.1093/imanum/draf030\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\\\\varOmega \\\\subset{\\\\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\\\\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\\\\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\\\\text{H}^{1}(\\\\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.\",\"PeriodicalId\":56295,\"journal\":{\"name\":\"IMA Journal of Numerical Analysis\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2025-06-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IMA Journal of Numerical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1093/imanum/draf030\",\"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":"IMA Journal of Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imanum/draf030","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Exponential convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction
We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\varOmega \subset{\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\text{H}^{1}(\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.
期刊介绍:
The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.