Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

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	Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises
其他题名	Well-posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises
	Yanzhao Cao 1; Jialin Hong 2; Zhihui Liu 3
	2020
发表期刊	数学研究通讯:英文版
ISSN	1674-5647
卷号	36.0 期号:002 页码:113-127
摘要	The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The noise term is approximated through the spectral projection of the covariance operator,which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises,the well-posedness of the SPDE is established under certain covariance operator-dependent conditions.These SPDEs with projected noises are then numerically approximated with the finite element method.A general error estimate framework is established for the finite element approximations.Based on this framework,optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained.It is shown that with the proposed approach,convergence order of white noise driven SPDEs is improved by half for one-dimensional problems,and by an infinitesimal factor for higher-dimensional problems.
其他摘要	The paper studies the well-posedness and optimal error estimates of spectralfinite element approximations for the boundary value problems of semi-linearelliptic SPDEs driven by white or colored Gaussian noises. The noise term isapproximated through the spectral projection of the covariance operator, whichis not required to be commutative with the Laplacian operator. Through theconvergence analysis of SPDEs with the noise terms replaced by the projectednoises, the well-posedness of the SPDE is established under certain covarianceoperator-dependent conditions. These SPDEs with projected noises are thennumerically approximated with the finite element method. A general errorestimate framework is established for the finite element approximations. Basedon this framework, optimal error estimates of finite element approximations forelliptic SPDEs driven by power-law noises are obtained. It is shown that withthe proposed approach, convergence order of white noise driven SPDEs isimproved by half for one-dimensional problems, and by an infinitesimal factorfor higher-dimensional problems.
关键词	Elliptic stochastic partial differential equation spectral approximations finite element approximations power-law noise
收录类别	CSCD
语种	中文
CSCD记录号	CSCD:6768199
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/57703
专题	中国科学院数学与系统科学研究院
作者单位	1.Department of Mathematics,Auburn University,Auburn,USA 2.中国科学院数学与系统科学研究院 3.Department of Mathematics,The Hong Kong University of Science and Technology,Hong Kong SAR
推荐引用方式 GB/T 7714	Yanzhao Cao,Jialin Hong,Zhihui Liu. Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises[J]. 数学研究通讯:英文版,2020,36.0(002):113-127.
APA	Yanzhao Cao,Jialin Hong,&Zhihui Liu.(2020).Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises.数学研究通讯:英文版,36.0(002),113-127.
MLA	Yanzhao Cao,et al."Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises".数学研究通讯:英文版 36.0.002(2020):113-127.