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

Sonja G. Cox, Kristin Kirchner
2020 Numerische Mathematik  
AbstractWe analyze several types of Galerkin approximations of a Gaussian random field $$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$ Z : D × Ω → R indexed by a Euclidean domain $$\mathscr {D}\subset \mathbb {R}^d$$ D ⊂ R d whose covariance structure is determined by a negative fractional power $$L^{-2\beta }$$ L - 2 β of a second-order elliptic differential operator $$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$ L : = - ∇ · ( A ∇ ) + κ 2 . Under minimal assumptions on the
more » ... omain $$\mathscr {D}$$ D , the coefficients $$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$ A : D → R d × d , $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R , and the fractional exponent $$\beta >0$$ β > 0 , we prove convergence in $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) and in $$L_q(\varOmega ; C^\delta (\overline{\mathscr {D}}))$$ L q ( Ω ; C δ ( D ¯ ) ) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on $$H^{1+\alpha }(\mathscr {D})$$ H 1 + α ( D ) -regularity of the differential operator L, 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 $$H^{\sigma ,\sigma }(\mathscr {D}\times \mathscr {D})$$ H σ , σ ( D × D ) , showing convergence which is more than twice as fast compared to the corresponding $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( 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 ≡ Id R d and $$\kappa \equiv {\text {const.}}$$ κ ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in $$d=2$$ d = 2 dimensions, where $$A:\mathscr {D}\rightarrow \mathbb {R}^{2\times 2}$$ A : D → R 2 × 2 and $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R are spatially varying.
doi:10.1007/s00211-020-01151-x fatcat:ihfke4espjgoplrl4wcscsyzzy