Year: 2020
Author: Yanzhao Cao, Jialin Hong, Zhihui Liu
Communications in Mathematical Research , Vol. 36 (2020), Iss. 2 : pp. 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0006
Communications in Mathematical Research , Vol. 36 (2020), Iss. 2 : pp. 113–127
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Elliptic stochastic partial differential equation spectral approximations finite element approximations power-law noise.
Author Details
-
Symplectic Integration of Stochastic Hamiltonian Systems
Infinite-Dimensional Stochastic Hamiltonian Systems
Hong, Jialin | Sun, Liying2022
https://doi.org/10.1007/978-981-19-7670-4_4 [Citations: 0] -
Numerical approximation and dynamics of periodic solution in distribution of stochastic differential equations
Zhou, Jinhui | Zou, Yongkui | Konarovskyi, Vitalii | Yang, XueNumerical Algorithms, Vol. (2025), Iss.
https://doi.org/10.1007/s11075-025-02008-w [Citations: 0] -
Finite difference methods for stochastic Helmholtz equation driven by white noise
Cui, Yanzhen | Tang, Shibing | Zhang, ChaoJournal of Computational and Applied Mathematics, Vol. 457 (2025), Iss. P.116286
https://doi.org/10.1016/j.cam.2024.116286 [Citations: 0] -
Numerical Analysis of Stabilizer‐Free Weak Galerkin Finite Element Method for Time‐Dependent Differential Equation Under Low Regularity
Liu, Xuan | Zou, Yongkui | Wang, Yiying | Zhou, Chenguang | Wang, HuiminNumerical Methods for Partial Differential Equations, Vol. 41 (2025), Iss. 1
https://doi.org/10.1002/num.23165 [Citations: 0] -
On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations
Xu, Chengqiang | Wang, Yibo | Cao, WanrongNumerical Methods for Partial Differential Equations, Vol. 40 (2024), Iss. 4
https://doi.org/10.1002/num.23094 [Citations: 0]