半线性椭圆型单反应边值问题hp-ILGFEM的指数收敛性

IF 2.4 2区 数学 Q1 MATHEMATICS, APPLIED
Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler
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引用次数: 0

摘要

研究了一类具有单项式反力和解析强迫的半线性椭圆边值模型问题在有界多边形$\varOmega \子集{\mathbb{R}}^{2}$中具有有限条直边的全显式数值逼近。特别地,我们分析了$hp$型迭代线性化伽辽金($hp$-ILG)解的收敛性。我们对$\varOmega $的正则、简单分区序列上的符合$hp$-有限元(FE) Galerkin离散进行收敛性分析,并对$\varOmega $的角进行几何网格细化,多项式度随着几何网格细化而增加。对于由ILG求解器生成的离散解序列,具有与精确的$hp$-FE Galerkin解的指数收敛一致的停止准则,我们证明了在$\text{H}^{1}(\varOmega)$中对边值问题的唯一弱解的指数收敛性。数值实验表明,从所提出的格式中得到的数值近似在自由度数量和计算复杂度方面具有指数收敛性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.
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来源期刊
IMA Journal of Numerical Analysis
IMA Journal of Numerical Analysis 数学-应用数学
CiteScore
5.30
自引率
4.80%
发文量
79
审稿时长
6-12 weeks
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.
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