Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises (Communications in Mathematical Research, 2022)
The paper studies the well-posedness and optimal error estimatesof spectral finite element approximations for the boundary value problems ofsemi-linear elliptic SPDEs driven by white or colored Gaussian noises. Thenoise term is approximated through the spectral projection of the covarianceoperator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced bythe projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projectednoises are then numerically approximated with the finite element method. Ageneral error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite elementapproximations for elliptic SPDEs driven by power-law noises are obtained. Itis shown that with the proposed approach, convergence order of white noisedriven SPDEs is improved by half for one-dimensional problems, and by aninfinitesimal factor for higher-dimensional problems.
Keywords: Elliptic stochastic partial differential equation; spectral approximations; finite element approximations; power-law noise.