The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in Rd is considered. The differential operator is given by the fractional power Lβ, β ∈ (0,1) of an integer-order elliptic differential operator L and is therefore nonlocal. Its inverse L-β is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator L-β is approximated by a weighted sum of nonfractional resolvents (I + exp(2 yℓ) L)-1 at certain quadrature nodes tj > 0. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L=κ2-Δ, κ> 0 with homogeneous Dirichlet boundary conditions on the unit cube (0,1)d in d=1,2,3 spatial dimensions for varying β ∈ (0,1) attest to the theoretical results.
Bibliographical noteFunding Information:
Swedish Research Council (2016-04187 and 2017-04274); Knut and Alice Wallenberg Foundation (KAW 20012.0067).
© 2018 Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.
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- finite element methods
- fractional operators
- Gaussian white noise
- Matérn covariances
- spatial statistics
- stochastic partial differential equations
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics