arrow
Volume 36, Issue 2
Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

Yanzhao Cao, Jialin Hong & Zhihui Liu

Commun. Math. Res., 36 (2020), pp. 113-127.

Published online: 2020-05

Export citation
  • Abstract

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.

  • AMS Subject Headings

60H35, 65M60, 60H15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CMR-36-113, author = {Cao , YanzhaoHong , Jialin and Liu , Zhihui}, title = {Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {2}, pages = {113--127}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0006}, url = {http://global-sci.org/intro/article_detail/cmr/16925.html} }
TY - JOUR T1 - Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises AU - Cao , Yanzhao AU - Hong , Jialin AU - Liu , Zhihui JO - Communications in Mathematical Research VL - 2 SP - 113 EP - 127 PY - 2020 DA - 2020/05 SN - 36 DO - http://doi.org/10.4208/cmr.2020-0006 UR - https://global-sci.org/intro/article_detail/cmr/16925.html KW - Elliptic stochastic partial differential equation, spectral approximations, finite element approximations, power-law noise. AB -

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.

Yanzhao Cao, Jialin Hong & Zhihui Liu. (2020). Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises. Communications in Mathematical Research . 36 (2). 113-127. doi:10.4208/cmr.2020-0006
Copy to clipboard
The citation has been copied to your clipboard