Regularity and convergence analysis in Sobolev and H"older spaces for generalized Whittle-Mat'ern fields

We analyze several Galerkin approximations of a Gaussian random field ZcolonD×Omega→ℝ indexed by a Euclidean domain Dsubsetℝd whose covariance structure is determined by a negative fractional power L-2β of a second-order elliptic differential operator L:= -nabla·(Anabla) + kappa2. Under minimal assumptions on the domain D, the coefficients AcolonD→ℝd× d, kappacolonD→ℝ, and the fractional exponent β>0, we prove convergence in Lq(Omega; Hσ(D)) and in Lq(Omega; Cδ(overlineD)) at (essentially) optimal rates for (i) spectral Galerkin methods and (ii) finite element approximations. Specifically, our analysis is solely based on H1+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 Hσ,σ(D×D), showing convergence which is more than twice as fast compared to the corresponding Lq(Omega; Hσ(D))-rate. For the well-known example of such Gaussian random fields, the original Whittle-Mat'ern class, where L=-Delta + kappa2 and kappa equiv operatornameconst., we perform several numerical experiments which validate our theoretical results.