## Abstract

We analyze several types of Galerkin approximations of a Gaussian random field Z: D× Ω→ R indexed by a Euclidean domain D⊂ R^{d} whose covariance structure is determined by a negative fractional power L^{-}^{2}^{β} of a second-order elliptic differential operator L: = - ∇ · (A∇) + κ^{2}. Under minimal assumptions on the domain D, the coefficients A: D→ R^{d}^{×}^{d}, κ: D→ R, and the fractional exponent β> 0 , we prove convergence in L_{q}(Ω; H^{σ}(D)) and in 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}^{+}^{α}(D) -regularity of the differential operator L, where 0 < α≤ 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L_{∞}(D× D) and in the mixed Sobolev space H^{σ}^{,}^{σ}(D× D) , showing convergence which is more than twice as fast compared to the corresponding 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≡IdRd and κ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in d= 2 dimensions, where A: D→ R^{2 × 2} and κ: D→ R are spatially varying.

Original language | English |
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Pages (from-to) | 819-873 |

Number of pages | 55 |

Journal | Numerische Mathematik |

Volume | 146 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2020 |