We analyze several Galerkin approximations of a Gaussian random field $mathcal{Z}colonmathcal{D}timesOmegatomathbb{R}$ indexed by a Euclidean domain $mathcal{D}subsetmathbb{R}^d$ whose covariance structure is determined by a negative fractional power $L^{-2beta}$ of a second-order elliptic differential operator $L:= - ablacdot(A abla) + kappa^2$. Under minimal assumptions on the domain $mathcal{D}$, the coefficients $Acolonmathcal{D}tomathbb{R}^{dtimes d}$, $kappacolonmathcal{D}tomathbb{R}$, and the fractional exponent $beta>0$, we prove convergence in $L_q(Omega; H^sigma(mathcal{D}))$ and in $L_q(Omega; C^delta(overline{mathcal{D}}))$ at (essentially) optimal rates for (i) spectral Galerkin methods and (ii) finite element approximations. Specifically, our analysis is solely based on $H^{1+alpha}(mathcal{D})$-regularity of the differential operator $L$, where $0<alphaleq 1$. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $L_{infty}(mathcal{D}timesmathcal{D})$ and in the mixed Sobolev space $H^{sigma,sigma}(mathcal{D}timesmathcal{D})$, showing convergence which is more than twice as fast compared to the corresponding $L_q(Omega; H^sigma(mathcal{D}))$-rate. For the well-known example of such Gaussian random fields, the original Whittle-Matern class, where $L=-Delta + kappa^2$ and $kappa equiv operatorname{const.}$, we perform several numerical experiments which validate our theoretical results.
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