The optimal convergence of finite element methods for the three-dimensional second-order elliptic boundary value problem with singularity

Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 f...

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Published inJournal of mathematical analysis and applications Vol. 491; no. 2; p. 124369
Main Authors He, Wen-ming, Zhao, Ren, Li, Guanrong, Xiao, Jin, Li, Wulan
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.11.2020
Subjects
Elliptic boundary value problem
Graded mesh
Sobolev space
Graded mesh
Sobolev space
Elliptic boundary value problem
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Summary:Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 for approximations with a Pk finite element method.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2020.124369