Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle-Matérn fields

We analyze several Galerkin approximations of a Gaussian random field $\mathcal{Z}\colon\mathcal{D}\timesΩ\to\mathbb{R}$ indexed by a Euclidean domain $\mathcal{D}\subset\mathbb{R}^d$ whose covariance structure is determined by a negative fractional power $L^{-2β}$ of a second-order elliptic differential operator $L:= -\nabla\cdot(A\nabla) + κ^2$. Under minimal assumptions on the domain $\mathcal{D}$, the coefficients $A\colon\mathcal{D}\to\mathbb{R}^{d\times d}$, $κ\colon\mathcal{D}\to\mathbb{R}$, and the fractional exponent $β>0$, we prove convergence in $L_q(Ω; H^σ(\mathcal{D}))$ and in $L_q(Ω; C^δ(\overline{\mathcal{D}}))$ at (essentially) optimal rates for (i) spectral Galerkin methods and (ii) finite element approximations. Specifically, our analysis is solely based on $H^{1+α}(\mathcal{D})$-regularity of the differential operator $L$, where $0<α\leq 1$. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $L_{\infty}(\mathcal{D}\times\mathcal{D})$ and in the mixed Sobolev space $H^{σ,σ}(\mathcal{D}\times\mathcal{D})$, showing convergence which is more than twice as fast compared to the corresponding $L_q(Ω; H^σ(\mathcal{D}))$-rate. For the well-known example of such Gaussian random fields, the original Whittle-Matérn class, where $L=-Δ+ κ^2$ and $κ\equiv \operatorname{const.}$, we perform several numerical experiments which validate our theoretical results.

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