  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.
