TY - JOUR

T1 - Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

AU - Bolin, David

AU - Kirchner, Kristin

AU - Kovács, Mihály

N1 - Funding Information:
Swedish Research Council (2016-04187 and 2017-04274); Knut and Alice Wallenberg Foundation (KAW 20012.0067).
Publisher Copyright:
© 2018 Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/4/24

Y1 - 2020/4/24

N2 - 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.

AB - 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.

KW - finite element methods

KW - fractional operators

KW - Gaussian white noise

KW - Matérn covariances

KW - spatial statistics

KW - stochastic partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=85089372634&partnerID=8YFLogxK

U2 - 10.1093/imanum/dry091

DO - 10.1093/imanum/dry091

M3 - Article

AN - SCOPUS:85089372634

SN - 0272-4979

VL - 40

SP - 1051

EP - 1073

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 2

ER -