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

Abstract

AbstractWe analyze several types of Galerkin approximations of a Gaussian random field$$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$Z:D×Ω→Rindexed by a Euclidean domain$$\mathscr {D}\subset \mathbb {R}^d$$D⊂Rdwhose covariance structure is determined by a negative fractional power$$L^{-2\beta }$$L-2βof a second-order elliptic differential operator$$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$L:=-∇·(A∇)+κ2. Under minimal assumptions on the domain $$\mathscr {D}$$D, the coefficients$$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$A:D→Rd×d,$$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$κ:D→R, and the fractional exponent$$\beta >0$$β>0, we prove convergence in$$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$Lq(Ω;Hσ(D))and in$$L_q(\varOmega ; C^\delta (\overline{\mathscr {D}}))$$Lq(Ω;Cδ(D¯))at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on$$H^{1+\alpha }(\mathscr {D})$$H1+α(D)-regularity of the differential operatorL, where$$0<\alpha \le 1$$0<α≤1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in$$L_{\infty }(\mathscr {D}\times \mathscr {D})$$L∞(D×D)and in the mixed Sobolev space$$H^{\sigma ,\sigma }(\mathscr {D}\times \mathscr {D})$$Hσ,σ(D×D), showing convergence which is more than twice as fast compared to the corresponding$$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$Lq(Ω;Hσ(D))-rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where$$A\equiv \mathrm {Id}_{\mathbb {R}^d}$$A≡IdRdand$$\kappa \equiv {\text {const.}}$$κ≡const., and (b) an example of anisotropic, non-stationary Gaussian random fields in$$d=2$$d=2dimensions, where$$A:\mathscr {D}\rightarrow \mathbb {R}^{2\times 2}$$A:D→R2×2and$$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$κ:D→Rare spatially varying.