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

TL;DR

This paper establishes well-posedness and derives optimal finite element error estimates for semi-linear elliptic SPDEs driven by Gaussian noises, improving convergence rates especially for white noise in multiple dimensions.

Contribution

It introduces a spectral projection approach for noise approximation that does not require commutativity with the Laplacian, and develops a general error estimate framework for finite element methods.

Findings

Convergence order of white noise driven SPDEs is improved by half in 1D.

Optimal error estimates are obtained for power-law noises.

Well-posedness is established under covariance-dependent conditions.

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.