@Article{CMR-36-2,
author = {Yanzhao, Cao and Jialin, Hong and Zhihui, Liu},
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 = {<p style="text-align: justify;">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.</p>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2020-0006},
url = {https://global-sci.com/article/81526/well-posedness-and-finite-element-approximations-for-elliptic-spdes-with-gaussian-noises}
}